diff --git a/src/sage/manifolds/differentiable/metric.py b/src/sage/manifolds/differentiable/metric.py
index 686af0d3505..445862d6795 100644
--- a/src/sage/manifolds/differentiable/metric.py
+++ b/src/sage/manifolds/differentiable/metric.py
@@ -647,7 +647,7 @@ def set(self, symbiform):
             raise TypeError("the argument must be a tensor field")
         if symbiform._tensor_type != (0,2):
             raise TypeError("the argument must be of tensor type (0,2)")
-        if symbiform._sym != [(0,1)]:
+        if symbiform._sym != ((0,1),):
             raise TypeError("the argument must be symmetric")
         if not symbiform._domain.is_subset(self._domain):
             raise TypeError("the symmetric bilinear form is not defined " +
@@ -2301,7 +2301,7 @@ def set(self, symbiform):
                             "values on a parallelizable domain")
         if symbiform._tensor_type != (0,2):
             raise TypeError("the argument must be of tensor type (0,2)")
-        if symbiform._sym != [(0,1)]:
+        if symbiform._sym != ((0,1),):
             raise TypeError("the argument must be symmetric")
         if symbiform._vmodule is not self._vmodule:
             raise TypeError("the symmetric bilinear form and the metric are " +
@@ -2781,7 +2781,7 @@ def set(self, symbiform):
             raise TypeError("the argument must be a tensor field")
         if symbiform._tensor_type != (0,2):
             raise TypeError("the argument must be of tensor type (0,2)")
-        if symbiform._sym != [(0,1)]:
+        if symbiform._sym != ((0,1),):
             raise TypeError("the argument must be symmetric")
         if not symbiform._domain.is_subset(self._domain):
             raise TypeError("the symmetric bilinear form is not defined " +
@@ -3019,7 +3019,7 @@ def set(self, symbiform):
                             "values on a parallelizable domain")
         if symbiform._tensor_type != (0,2):
             raise TypeError("the argument must be of tensor type (0,2)")
-        if symbiform._sym != [(0,1)]:
+        if symbiform._sym != ((0,1),):
             raise TypeError("the argument must be symmetric")
         if symbiform._vmodule is not self._vmodule:
             raise TypeError("the symmetric bilinear form and the metric are " +
diff --git a/src/sage/manifolds/differentiable/pseudo_riemannian_submanifold.py b/src/sage/manifolds/differentiable/pseudo_riemannian_submanifold.py
index 2eaef7510c8..3aef31eb703 100644
--- a/src/sage/manifolds/differentiable/pseudo_riemannian_submanifold.py
+++ b/src/sage/manifolds/differentiable/pseudo_riemannian_submanifold.py
@@ -1173,7 +1173,7 @@ def ambient_second_fundamental_form(self):
                         g.restrict(chart.domain()).contract(pf[j]) *
                         self.scalar_field({chart: k.comp(chart.frame())[:][i, j]})
                         for i in range(self._dim) for j in range(self._dim))
-                    gam_rst._sym = [(0, 1)]
+                    gam_rst._sym = ((0, 1),)
                     self._ambient_second_fundamental_form.set_restriction(gam_rst)
 
                 charts = iter(self.top_charts())
diff --git a/src/sage/manifolds/differentiable/tensorfield.py b/src/sage/manifolds/differentiable/tensorfield.py
index d5b63c9d81d..a2aad4d4937 100644
--- a/src/sage/manifolds/differentiable/tensorfield.py
+++ b/src/sage/manifolds/differentiable/tensorfield.py
@@ -58,6 +58,7 @@
 from sage.rings.integer import Integer
 from sage.rings.integer_ring import ZZ
 from sage.structure.element import ModuleElementWithMutability
+from sage.tensor.modules.comp import CompWithSym
 from sage.tensor.modules.free_module_tensor import FreeModuleTensor
 from sage.tensor.modules.tensor_with_indices import TensorWithIndices
 
@@ -495,40 +496,8 @@ def __init__(
         self._restrictions = {} # dict. of restrictions of self on subdomains
                                 # of self._domain, with the subdomains as keys
         # Treatment of symmetry declarations:
-        self._sym = []
-        if sym is not None and sym != []:
-            if isinstance(sym[0], (int, Integer)):
-                # a single symmetry is provided as a tuple -> 1-item list:
-                sym = [tuple(sym)]
-            for isym in sym:
-                if len(isym) > 1:
-                    for i in isym:
-                        if i < 0 or i > self._tensor_rank - 1:
-                            raise IndexError("invalid position: {}".format(i) +
-                                 " not in [0,{}]".format(self._tensor_rank-1))
-                    self._sym.append(tuple(isym))
-        self._antisym = []
-        if antisym is not None and antisym != []:
-            if isinstance(antisym[0], (int, Integer)):
-                # a single antisymmetry is provided as a tuple -> 1-item list:
-                antisym = [tuple(antisym)]
-            for isym in antisym:
-                if len(isym) > 1:
-                    for i in isym:
-                        if i < 0 or i > self._tensor_rank - 1:
-                            raise IndexError("invalid position: {}".format(i) +
-                                " not in [0,{}]".format(self._tensor_rank-1))
-                    self._antisym.append(tuple(isym))
-        # Final consistency check:
-        index_list = []
-        for isym in self._sym:
-            index_list += isym
-        for isym in self._antisym:
-            index_list += isym
-        if len(index_list) != len(set(index_list)):
-            # There is a repeated index position:
-            raise IndexError("incompatible lists of symmetries: the same " +
-                             "position appears more than once")
+        self._sym, self._antisym = CompWithSym._canonicalize_sym_antisym(
+            self._tensor_rank, sym, antisym)
         # Initialization of derived quantities:
         self._init_derived()
 
@@ -591,7 +560,7 @@ def _repr_(self):
 
         """
         # Special cases
-        if self._tensor_type == (0,2) and self._sym == [(0,1)]:
+        if self._tensor_type == (0,2) and self._sym == ((0,1),):
             description = "Field of symmetric bilinear forms "
             if self._name is not None:
                 description += self._name + " "
@@ -957,13 +926,13 @@ def symmetries(self):
         elif len(self._sym) == 1:
             s = "symmetry: {}; ".format(self._sym[0])
         else:
-            s = "symmetries: {}; ".format(self._sym)
+            s = "symmetries: {}; ".format(list(self._sym))
         if not self._antisym:
             a = "no antisymmetry"
         elif len(self._antisym) == 1:
             a = "antisymmetry: {}".format(self._antisym[0])
         else:
-            a = "antisymmetries: {}".format(self._antisym)
+            a = "antisymmetries: {}".format(list(self._antisym))
         print(s + a)
 
     #### End of simple accessors #####
diff --git a/src/sage/manifolds/differentiable/vectorfield_module.py b/src/sage/manifolds/differentiable/vectorfield_module.py
index cf3f1dc687d..393218bc7c0 100644
--- a/src/sage/manifolds/differentiable/vectorfield_module.py
+++ b/src/sage/manifolds/differentiable/vectorfield_module.py
@@ -787,7 +787,7 @@ def tensor(self, tensor_type, name=None, latex_name=None, sym=None,
                 # a single antisymmetry is provided as a tuple or a
                 # range object; it is converted to a 1-item list:
                 antisym = [tuple(antisym)]
-            if isinstance(antisym, list):
+            if isinstance(antisym, (tuple, list)):
                 antisym0 = antisym[0]
             else:
                 antisym0 = antisym
@@ -799,7 +799,7 @@ def tensor(self, tensor_type, name=None, latex_name=None, sym=None,
                 # a single antisymmetry is provided as a tuple or a
                 # range object; it is converted to a 1-item list:
                 antisym = [tuple(antisym)]
-            if isinstance(antisym, list):
+            if isinstance(antisym, (tuple, list)):
                 antisym0 = antisym[0]
             else:
                 antisym0 = antisym
@@ -2102,7 +2102,7 @@ def tensor(self, tensor_type, name=None, latex_name=None, sym=None,
                 # a single antisymmetry is provided as a tuple or a
                 # range object; it is converted to a 1-item list:
                 antisym = [tuple(antisym)]
-            if isinstance(antisym, list):
+            if isinstance(antisym, (tuple, list)):
                 antisym0 = antisym[0]
             else:
                 antisym0 = antisym
@@ -2114,7 +2114,7 @@ def tensor(self, tensor_type, name=None, latex_name=None, sym=None,
                 # a single antisymmetry is provided as a tuple or a
                 # range object; it is converted to a 1-item list:
                 antisym = [tuple(antisym)]
-            if isinstance(antisym, list):
+            if isinstance(antisym, (tuple, list)):
                 antisym0 = antisym[0]
             else:
                 antisym0 = antisym
diff --git a/src/sage/tensor/modules/comp.py b/src/sage/tensor/modules/comp.py
index 38ee9c58601..3a33e49efd8 100644
--- a/src/sage/tensor/modules/comp.py
+++ b/src/sage/tensor/modules/comp.py
@@ -1806,12 +1806,12 @@ def __mul__(self, other):
                              "same starting index")
         if isinstance(other, CompWithSym):
             sym = []
-            if other._sym != []:
+            if other._sym:
                 for s in other._sym:
                     ns = tuple(s[i]+self._nid for i in range(len(s)))
                     sym.append(ns)
             antisym = []
-            if other._antisym != []:
+            if other._antisym:
                 for s in other._antisym:
                     ns = tuple(s[i]+self._nid for i in range(len(s)))
                     antisym.append(ns)
@@ -3029,46 +3029,76 @@ def __init__(self, ring, frame, nb_indices, start_index=0,
         """
         Components.__init__(self, ring, frame, nb_indices, start_index,
                             output_formatter)
-        self._sym = []
-        if sym is not None and sym != []:
+        self._sym, self._antisym = self._canonicalize_sym_antisym(
+            nb_indices, sym, antisym)
+
+    @staticmethod
+    def _canonicalize_sym_antisym(nb_indices, sym=None, antisym=None):
+        r"""
+        Bring sym and antisym into their canonical form.
+
+        EXAMPLES::
+
+            sage: from sage.tensor.modules.comp import CompWithSym
+            sage: CompWithSym._canonicalize_sym_antisym(6, [(2, 1)])
+            (((1, 2),), ())
+        """
+        result_sym = []
+        if sym is None:
+            sym = []
+        else:
+            # Handle the case that sym is an iterator
+            sym = list(sym)
+        if sym:
             if isinstance(sym[0], (int, Integer)):
                 # a single symmetry is provided as a tuple or a range object;
                 # it is converted to a 1-item list:
                 sym = [tuple(sym)]
             for isym in sym:
                 if len(isym) < 2:
-                    raise IndexError("at least two index positions must be " +
-                                     "provided to define a symmetry")
+                    # Drop trivial symmetry
+                    continue
                 for i in isym:
-                    if i<0 or i>self._nid-1:
+                    if i < 0 or i > nb_indices - 1:
                         raise IndexError("invalid index position: " + str(i) +
-                                         " not in [0," + str(self._nid-1) + "]")
-                self._sym.append(tuple(isym))
-        self._antisym = []
-        if antisym is not None and antisym != []:
+                                         " not in [0," + str(nb_indices-1) + "]")
+                result_sym.append(tuple(isym))
+        result_antisym = []
+        if antisym is None:
+            antisym = []
+        else:
+            # Handle the case that antisym is an iterator
+            antisym = list(antisym)
+        if antisym:
             if isinstance(antisym[0], (int, Integer)):
                 # a single antisymmetry is provided as a tuple or a range
                 # object; it is converted to a 1-item list:
                 antisym = [tuple(antisym)]
             for isym in antisym:
                 if len(isym) < 2:
-                    raise IndexError("at least two index positions must be " +
-                                     "provided to define an antisymmetry")
+                    # Drop trivial antisymmetry
+                    continue
                 for i in isym:
-                    if i<0 or i>self._nid-1:
+                    if i < 0 or i > nb_indices - 1:
                         raise IndexError("invalid index position: " + str(i) +
-                                         " not in [0," + str(self._nid-1) + "]")
-                self._antisym.append(tuple(isym))
+                                         " not in [0," + str(nb_indices - 1) + "]")
+                result_antisym.append(tuple(isym))
         # Final consistency check:
         index_list = []
-        for isym in self._sym:
+        for isym in result_sym:
             index_list += isym
-        for isym in self._antisym:
+        for isym in result_antisym:
             index_list += isym
         if len(index_list) != len(set(index_list)):
             # There is a repeated index position:
             raise IndexError("incompatible lists of symmetries: the same " +
-                             "index position appears more then once")
+                             "index position appears more than once")
+        # Canonicalize sort order, make tuples
+        result_sym = [tuple(sorted(s)) for s in result_sym]
+        result_antisym = [tuple(sorted(s)) for s in result_antisym]
+        result_sym = tuple(sorted(result_sym))
+        result_antisym = tuple(sorted(result_antisym))
+        return result_sym, result_antisym
 
     def _repr_symmetry(self):
         r"""
@@ -3394,9 +3424,9 @@ def swap_adjacent_indices(self, pos1, pos2, pos3):
              [[0, 7, 8], [-7, 0, 9], [-8, -9, 0]]]
             sage: c1 = c.swap_adjacent_indices(0,1,3)
             sage: c._antisym   # c is antisymmetric with respect to the last pair of indices...
-            [(1, 2)]
+            ((1, 2),)
             sage: c1._antisym  #...while c1 is antisymmetric with respect to the first pair of indices
-            [(0, 1)]
+            ((0, 1),)
             sage: c[0,1,2]
             3
             sage: c1[1,2,0]
@@ -3417,6 +3447,8 @@ def swap_adjacent_indices(self, pos1, pos2, pos3):
         for s in self._antisym:
             new_s = [new_lpos.index(pos) for pos in s]
             result._antisym.append(tuple(sorted(new_s)))
+        result._sym, result._antisym = self._canonicalize_sym_antisym(
+            self._nid, result._sym, result._antisym)
         # The values:
         for ind, val in self._comp.items():
             new_ind = ind[:pos1] + ind[pos2:pos3] + ind[pos1:pos2] + ind[pos3:]
@@ -3547,7 +3579,7 @@ def paral_sum(a, b, local_list_ind):
                         com = tuple(set(isym).intersection(set(osym)))
                         if len(com) > 1:
                             common_antisym.append(com)
-                if common_sym != [] or common_antisym != []:
+                if common_sym or common_antisym:
                     result = CompWithSym(self._ring, self._frame, self._nid,
                                          self._sindex, self._output_formatter,
                                          common_sym, common_antisym)
@@ -3655,11 +3687,11 @@ def __mul__(self, other):
         sym = list(self._sym)
         antisym = list(self._antisym)
         if isinstance(other, CompWithSym):
-            if other._sym != []:
+            if other._sym:
                 for s in other._sym:
                     ns = tuple(s[i]+self._nid for i in range(len(s)))
                     sym.append(ns)
-            if other._antisym != []:
+            if other._antisym:
                 for s in other._antisym:
                     ns = tuple(s[i]+self._nid for i in range(len(s)))
                     antisym.append(ns)
@@ -3985,7 +4017,7 @@ def non_redundant_index_generator(self):
         si = self._sindex
         imax = self._dim - 1 + si
         ind = [si for k in range(self._nid)]
-        sym = self._sym.copy() # we may modify this in the following
+        sym = list(self._sym)  # we may modify this in the following
         antisym = self._antisym
         for pos in range(self._nid):
             for isym in antisym:
@@ -4161,7 +4193,7 @@ def symmetrize(self, *pos):
             (0, 0, 1)
             ], with symmetry on the index positions (0, 1), with symmetry on the index positions (2, 3)
             sage: a1._sym  # a1 has two distinct symmetries:
-            [(0, 1), (2, 3)]
+            ((0, 1), (2, 3))
             sage: a[0,1,2,0] == a[0,0,2,1]  # a is symmetric w.r.t. positions 1 and 3
             True
             sage: a1[0,1,2,0] == a1[0,0,2,1] # a1 is not
@@ -4439,10 +4471,10 @@ def antisymmetrize(self, *pos):
             (1, 0, 0),
             (0, 1, 0),
             (0, 0, 1)
-            ], with antisymmetry on the index positions (1, 3),
-               with antisymmetry on the index positions (0, 2)
+            ], with antisymmetry on the index positions (0, 2),
+               with antisymmetry on the index positions (1, 3)
             sage: s._antisym  # the antisymmetry (0,1,2) has been reduced to (0,2), since 1 is involved in the new antisymmetry (1,3):
-            [(1, 3), (0, 2)]
+            ((0, 2), (1, 3))
 
         Partial antisymmetrization of 4-indices components with a symmetry on
         the first two indices::
diff --git a/src/sage/tensor/modules/finite_rank_free_module.py b/src/sage/tensor/modules/finite_rank_free_module.py
index d746f922309..a08b17b1129 100644
--- a/src/sage/tensor/modules/finite_rank_free_module.py
+++ b/src/sage/tensor/modules/finite_rank_free_module.py
@@ -704,9 +704,8 @@ def ambient_module(self): # compatible with sage.modules.free_module.FreeModule_
             sage: M.ambient_module() is M
             True
 
-            sage: from sage.tensor.modules.tensor_free_submodule import TensorFreeSubmodule_comp
             sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
-            sage: Sym0123x45M = TensorFreeSubmodule_comp(M, (6, 0), sym=((0, 1, 2, 3), (4, 5)))
+            sage: Sym0123x45M = M.tensor_module(6, 0, sym=((0, 1, 2, 3), (4, 5)))
             sage: T60M = M.tensor_module(6, 0)
             sage: Sym0123x45M.ambient_module() is T60M
             True
@@ -1429,12 +1428,19 @@ def tensor_module(self, k, l, *, sym=None, antisym=None):
             T^{\{2,3\}}(M) \otimes T^{\{6,7\}}(M^*) \otimes \mathrm{Sym}^{\{0,1\}}(M) \otimes \mathrm{ASym}^{\{4,5\}}(M^*)
 
         See :class:`~sage.tensor.modules.tensor_free_module.TensorFreeModule`
-        and :class:`~sage.tensor.modules.tensor_free_module.TensorFreeSubmodule_comp`
+        and :class:`~sage.tensor.modules.tensor_free_module.TensorFreeSubmodule_sym`
         for more documentation.
 
+        TESTS::
+
+            sage: M = FiniteRankFreeModule(ZZ, 2)
+            sage: M.tensor_module(2, 0, sym=(0,1)) is M.symmetric_power(2)
+            True
         """
+        from .comp import CompWithSym, CompFullyAntiSym
+
+        sym, antisym = CompWithSym._canonicalize_sym_antisym(k + l, sym, antisym)
         if sym or antisym:
-            # TODO: Canonicalize sym, antisym, make hashable
             key = (k, l, sym, antisym)
         else:
             key = (k, l)
@@ -1444,8 +1450,8 @@ def tensor_module(self, k, l, *, sym=None, antisym=None):
             if key == (1, 0):
                 T = self
             elif sym or antisym:
-                from sage.tensor.modules.tensor_free_submodule import TensorFreeSubmodule_comp
-                T = TensorFreeSubmodule_comp(self, (k, l), sym=sym, antisym=antisym)
+                from sage.tensor.modules.tensor_free_submodule import TensorFreeSubmodule_sym
+                T = TensorFreeSubmodule_sym(self, (k, l), sym=sym, antisym=antisym)
             else:
                 from sage.tensor.modules.tensor_free_module import TensorFreeModule
                 T = TensorFreeModule(self, (k, l))
@@ -2031,7 +2037,7 @@ def tensor(self, tensor_type, name=None, latex_name=None, sym=None,
                 # a single antisymmetry is provided as a tuple or a range
                 # object; it is converted to a 1-item list:
                 antisym = [tuple(antisym)]
-            if isinstance(antisym, list):
+            if isinstance(antisym, (tuple, list)):
                 antisym0 = antisym[0]
             else:
                 antisym0 = antisym
@@ -2043,7 +2049,7 @@ def tensor(self, tensor_type, name=None, latex_name=None, sym=None,
                 # a single antisymmetry is provided as a tuple or a range
                 # object; it is converted to a 1-item list:
                 antisym = [tuple(antisym)]
-            if isinstance(antisym, list):
+            if isinstance(antisym, (tuple, list)):
                 antisym0 = antisym[0]
             else:
                 antisym0 = antisym
diff --git a/src/sage/tensor/modules/free_module_tensor.py b/src/sage/tensor/modules/free_module_tensor.py
index 845c9185a26..2ffa00cc476 100644
--- a/src/sage/tensor/modules/free_module_tensor.py
+++ b/src/sage/tensor/modules/free_module_tensor.py
@@ -309,41 +309,8 @@ def __init__(
                               # bases, with the bases as keys (initially empty)
 
         # Treatment of symmetry declarations:
-        self._sym = []
-        if sym is not None and sym != []:
-            if isinstance(sym[0], (int, Integer)):
-                # a single symmetry is provided as a tuple -> 1-item list:
-                sym = [tuple(sym)]
-            for isym in sym:
-                if len(isym) > 1:
-                    for i in isym:
-                        if i<0 or i>self._tensor_rank-1:
-                            raise IndexError("invalid position: " + str(i) +
-                                " not in [0," + str(self._tensor_rank-1) + "]")
-                    self._sym.append(tuple(isym))
-        self._antisym = []
-        if antisym is not None and antisym != []:
-            if isinstance(antisym[0], (int, Integer)):
-                # a single antisymmetry is provided as a tuple -> 1-item list:
-                antisym = [tuple(antisym)]
-            for isym in antisym:
-                if len(isym) > 1:
-                    for i in isym:
-                        if i<0 or i>self._tensor_rank-1:
-                            raise IndexError("invalid position: " + str(i) +
-                                " not in [0," + str(self._tensor_rank-1) + "]")
-                    self._antisym.append(tuple(isym))
-
-        # Final consistency check:
-        index_list = []
-        for isym in self._sym:
-            index_list += isym
-        for isym in self._antisym:
-            index_list += isym
-        if len(index_list) != len(set(index_list)):
-            # There is a repeated index position:
-            raise IndexError("incompatible lists of symmetries: the same " +
-                             "position appears more than once")
+        self._sym, self._antisym = CompWithSym._canonicalize_sym_antisym(
+            self._tensor_rank, sym, antisym)
 
         # Initialization of derived quantities:
         FreeModuleTensor._init_derived(self)
@@ -405,7 +372,7 @@ def _repr_(self):
 
         """
         # Special cases
-        if self._tensor_type == (0,2) and self._sym == [(0,1)]:
+        if self._tensor_type == (0,2) and self._sym == ((0,1),):
             description = "Symmetric bilinear form "
         else:
             # Generic case
@@ -563,13 +530,13 @@ def symmetries(self):
         elif len(self._sym) == 1:
             s = "symmetry: {}; ".format(self._sym[0])
         else:
-            s = "symmetries: {}; ".format(self._sym)
+            s = "symmetries: {}; ".format(list(self._sym))
         if len(self._antisym) == 0:
             a = "no antisymmetry"
         elif len(self._antisym) == 1:
             a = "antisymmetry: {}".format(self._antisym[0])
         else:
-            a = "antisymmetries: {}".format(self._antisym)
+            a = "antisymmetries: {}".format(list(self._antisym))
         print(s+a)
 
     #### End of simple accessors #####
diff --git a/src/sage/tensor/modules/tensor_free_module.py b/src/sage/tensor/modules/tensor_free_module.py
index d4f000d7be6..42a97d92b4b 100644
--- a/src/sage/tensor/modules/tensor_free_module.py
+++ b/src/sage/tensor/modules/tensor_free_module.py
@@ -66,7 +66,7 @@
 from sage.tensor.modules.free_module_morphism import \
                                                    FiniteRankFreeModuleMorphism
 from sage.tensor.modules.free_module_automorphism import FreeModuleAutomorphism
-from .tensor_free_submodule_basis import TensorFreeSubmoduleBasis_comp
+from .tensor_free_submodule_basis import TensorFreeSubmoduleBasis_sym
 
 class TensorFreeModule(FiniteRankFreeModule_abstract):
     r"""
@@ -554,11 +554,10 @@ def _an_element_(self):
 
         TESTS::
 
-            sage: from sage.tensor.modules.tensor_free_submodule import TensorFreeSubmodule_comp
             sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
             sage: e = M.basis('e')
             sage: T60M = M.tensor_module(6, 0)
-            sage: Sym0123x45M = TensorFreeSubmodule_comp(M, (6, 0), sym=((0, 1, 2, 3), (4, 5)))
+            sage: Sym0123x45M = M.tensor_module(6, 0, sym=((0, 1, 2, 3), (4, 5)))
             sage: t = Sym0123x45M._an_element_()
             sage: t.parent() is Sym0123x45M
             True
@@ -623,8 +622,7 @@ def _coerce_map_from_(self, other):
 
         Coercion from submodules::
 
-            sage: from sage.tensor.modules.tensor_free_submodule import TensorFreeSubmodule_comp
-            sage: Sym01M = TensorFreeSubmodule_comp(M, (2, 0), sym=((0, 1)))
+            sage: Sym01M = M.tensor_module(2, 0, sym=((0, 1)))
             sage: M.tensor_module(2,0)._coerce_map_from_(Sym01M)
             True
 
@@ -732,7 +730,7 @@ def basis(self, symbol, latex_symbol=None, from_family=None,
 
         INPUT:
 
-        - ``symbol``, ``indices`` -- passed to he base module's method
+        - ``symbol``, ``indices`` -- passed to the base module's method
           :meth:`~sage.tensor.modules.finite_rank_free_module.FiniteRankFreeModule.basis`
           to select a basis of the :meth:`base_module` of ``self``,
           or to create it.
@@ -760,8 +758,7 @@ def basis(self, symbol, latex_symbol=None, from_family=None,
             e_2⊗e^1
             e_2⊗e^2
 
-            sage: from sage.tensor.modules.tensor_free_submodule import TensorFreeSubmodule_comp
-            sage: Sym2M = TensorFreeSubmodule_comp(M, (2, 0), sym=range(2))
+            sage: Sym2M = M.tensor_module(2, 0, sym=range(2))
             sage: e_Sym2M = Sym2M.basis('e'); e_Sym2M
             Standard basis on the
              Free module of fully symmetric type-(2,0) tensors on the Rank-3 free module M over the Integer Ring
@@ -773,26 +770,44 @@ def basis(self, symbol, latex_symbol=None, from_family=None,
             e_1⊗e_1
             e_1⊗e_2 + e_2⊗e_1
             e_2⊗e_2
+
+            sage: M = FiniteRankFreeModule(ZZ, 2)
+            sage: e = M.basis('e')
+            sage: f = M.basis('f', from_family=(-e[1], e[0]))
+            sage: for b in f: b.display()
+            f_0 = -e_1
+            f_1 = e_0
+            sage: S = M.tensor_module(2, 0, sym=(0,1))
+            sage: fS = S.basis('f')
+            sage: for b in fS: b.display()
+            e_1⊗e_1
+            -e_0⊗e_1 - e_1⊗e_0
+            e_0⊗e_0
+            sage: for b in fS: b.display(f)
+            f_0⊗f_0
+            f_0⊗f_1 + f_1⊗f_0
+            f_1⊗f_1
+
         """
-        return TensorFreeSubmoduleBasis_comp(self, symbol=symbol, latex_symbol=latex_symbol,
+        return TensorFreeSubmoduleBasis_sym(self, symbol=symbol, latex_symbol=latex_symbol,
                                              indices=indices, latex_indices=latex_indices,
                                              symbol_dual=symbol_dual, latex_symbol_dual=latex_symbol_dual)
 
     @cached_method
-    def _basis_comp(self):
+    def _basis_sym(self):
         r"""
         Return an instance of :class:`~sage.tensor.modules.comp.Components`.
 
         This implementation returns an instance without symmetry.
 
-        The subclass :class:`~sage.tensor.modules.tensor_free_submodule.TensorFreeSubmodule_comp`
+        The subclass :class:`~sage.tensor.modules.tensor_free_submodule.TensorFreeSubmodule_sym`
         overrides this method to encode the prescribed symmetry of the submodule.
 
         EXAMPLES::
 
             sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
             sage: T = M.tensor_module(1,1)
-            sage: c = T._basis_comp(); c
+            sage: c = T._basis_sym(); c
             2-indices components w.r.t. (0, 1, 2)
 
         """
diff --git a/src/sage/tensor/modules/tensor_free_submodule.py b/src/sage/tensor/modules/tensor_free_submodule.py
index dc94ace3343..cfe6b09dccf 100644
--- a/src/sage/tensor/modules/tensor_free_submodule.py
+++ b/src/sage/tensor/modules/tensor_free_submodule.py
@@ -1,5 +1,5 @@
 r"""
-Free submodules of tensor products of free modules
+Free submodules of tensor modules defined by monoterm symmetries
 """
 
 #******************************************************************************
@@ -23,15 +23,13 @@
 from .finite_rank_free_module import FiniteRankFreeModule_abstract
 
 
-class TensorFreeSubmodule_comp(TensorFreeModule):
+class TensorFreeSubmodule_sym(TensorFreeModule):
     r"""
     Class for free submodules of tensor products of free modules
-    that are defined by the symmetries of a
-    :class:`~sage.tensor.modules.comp.Components` object.
+    that are defined by some monoterm symmetries.
 
     EXAMPLES::
 
-        sage: from sage.tensor.modules.tensor_free_submodule import TensorFreeSubmodule_comp
         sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
         sage: e = M.basis('e')
         sage: T60M = M.tensor_module(6, 0); T60M
@@ -40,14 +38,14 @@ class TensorFreeSubmodule_comp(TensorFreeModule):
         'T^(6, 0)(M)'
         sage: latex(T60M)
         T^{(6, 0)}\left(M\right)
-        sage: T40Sym45M = TensorFreeSubmodule_comp(M, (6, 0), sym=((4, 5))); T40Sym45M
+        sage: T40Sym45M = M.tensor_module(6, 0, sym=((4, 5))); T40Sym45M
         Free module of type-(6,0) tensors on the Rank-3 free module M over the Integer Ring,
          with symmetry on the index positions (4, 5)
         sage: T40Sym45M._name
         'T^{0,1,2,3}(M)⊗Sym^{4,5}(M)'
         sage: latex(T40Sym45M)
         T^{\{0,1,2,3\}}(M) \otimes \mathrm{Sym}^{\{4,5\}}(M)
-        sage: Sym0123x45M = TensorFreeSubmodule_comp(M, (6, 0), sym=((0, 1, 2, 3), (4, 5))); Sym0123x45M
+        sage: Sym0123x45M = M.tensor_module(6, 0, sym=((0, 1, 2, 3), (4, 5))); Sym0123x45M
         Free module of type-(6,0) tensors on the Rank-3 free module M over the Integer Ring,
          with symmetry on the index positions (0, 1, 2, 3),
          with symmetry on the index positions (4, 5)
@@ -55,7 +53,7 @@ class TensorFreeSubmodule_comp(TensorFreeModule):
         'Sym^{0,1,2,3}(M)⊗Sym^{4,5}(M)'
         sage: latex(Sym0123x45M)
         \mathrm{Sym}^{\{0,1,2,3\}}(M) \otimes \mathrm{Sym}^{\{4,5\}}(M)
-        sage: Sym012x345M = TensorFreeSubmodule_comp(M, (6, 0), sym=((0, 1, 2), (3, 4, 5))); Sym012x345M
+        sage: Sym012x345M = M.tensor_module(6, 0, sym=((0, 1, 2), (3, 4, 5))); Sym012x345M
         Free module of type-(6,0) tensors on the Rank-3 free module M over the Integer Ring,
          with symmetry on the index positions (0, 1, 2),
          with symmetry on the index positions (3, 4, 5)
@@ -63,7 +61,7 @@ class TensorFreeSubmodule_comp(TensorFreeModule):
         'Sym^{0,1,2}(M)⊗Sym^{3,4,5}(M)'
         sage: latex(Sym012x345M)
         \mathrm{Sym}^{\{0,1,2\}}(M) \otimes \mathrm{Sym}^{\{3,4,5\}}(M)
-        sage: Sym012345M = TensorFreeSubmodule_comp(M, (6, 0), sym=((0, 1, 2, 3, 4, 5))); Sym012345M
+        sage: Sym012345M = M.tensor_module(6, 0, sym=((0, 1, 2, 3, 4, 5))); Sym012345M
         Free module of fully symmetric type-(6,0) tensors
          on the Rank-3 free module M over the Integer Ring
         sage: Sym012345M._name
@@ -85,7 +83,7 @@ class TensorFreeSubmodule_comp(TensorFreeModule):
 
     TESTS::
 
-        sage: T = TensorFreeSubmodule_comp(M, (4, 4), sym=((0, 1)), antisym=((4, 5))); T
+        sage: T = M.tensor_module(4, 4, sym=((0, 1)), antisym=((4, 5))); T
         Free module of type-(4,4) tensors on the Rank-3 free module M over the Integer Ring,
          with symmetry on the index positions (0, 1),
          with antisymmetry on the index positions (4, 5)
@@ -100,10 +98,9 @@ def __init__(self, fmodule, tensor_type, name=None, latex_name=None,
         r"""
         TESTS::
 
-            sage: from sage.tensor.modules.tensor_free_submodule import TensorFreeSubmodule_comp
             sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
             sage: e = M.basis('e')
-            sage: Sym0123x45M = TensorFreeSubmodule_comp(M, (6, 0), sym=((0, 1, 2, 3), (4, 5)))
+            sage: Sym0123x45M = M.tensor_module(6, 0, sym=((0, 1, 2, 3), (4, 5)))
             sage: TestSuite(Sym0123x45M).run()
         """
         self._fmodule = fmodule
@@ -113,20 +110,20 @@ def __init__(self, fmodule, tensor_type, name=None, latex_name=None,
         self._ambient_module = ambient
         self._sym = sym
         self._antisym = antisym
-        basis_comp = self._basis_comp()
-        rank = len(list(basis_comp.non_redundant_index_generator()))
+        basis_sym = self._basis_sym()
+        rank = len(list(basis_sym.non_redundant_index_generator()))
 
         if name is None and fmodule._name is not None:
             all_indices = tuple(range(tensor_type[0] + tensor_type[1]))
-            if isinstance(basis_comp, CompFullySym):
+            if isinstance(basis_sym, CompFullySym):
                 sym = [all_indices]
                 antisym = []
-            elif isinstance(basis_comp, CompFullyAntiSym):
+            elif isinstance(basis_sym, CompFullyAntiSym):
                 sym = []
                 antisym = [all_indices]
-            elif isinstance(basis_comp, CompWithSym):
-                sym = basis_comp._sym
-                antisym = basis_comp._antisym
+            elif isinstance(basis_sym, CompWithSym):
+                sym = basis_sym._sym
+                antisym = basis_sym._antisym
             else:
                 sym = antisym = []
             nosym_0 = [i for i in range(tensor_type[0])
@@ -178,19 +175,19 @@ def power_name(op, s, latex=False):
                                                category=category, ambient=ambient)
 
     @cached_method
-    def _basis_comp(self):
+    def _basis_sym(self):
         r"""
         Return an instance of :class:`~sage.tensor.modules.comp.Components`.
 
-        It encodes the prescribed symmetry of ``self``.
+        In the current implementation of :class:`~sage.tensor.modules.tensor_free_submodule.TensorFreeSubmodule_sym`,
+        it encodes the prescribed symmetry of ``self``.
 
         EXAMPLES::
 
-            sage: from sage.tensor.modules.tensor_free_submodule import TensorFreeSubmodule_comp
             sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
-            sage: Sym2M = TensorFreeSubmodule_comp(M, (2, 0), sym=range(2)); Sym2M
+            sage: Sym2M = M.tensor_module(2, 0, sym=range(2)); Sym2M
             Free module of fully symmetric type-(2,0) tensors on the Rank-3 free module M over the Integer Ring
-            sage: c = Sym2M._basis_comp(); c
+            sage: c = Sym2M._basis_sym(); c
             Fully symmetric 2-indices components w.r.t. (0, 1, 2)
 
         """
@@ -205,14 +202,13 @@ def _repr_(self):
 
         EXAMPLES::
 
-            sage: from sage.tensor.modules.tensor_free_submodule import TensorFreeSubmodule_comp
             sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
-            sage: Sym2M = TensorFreeSubmodule_comp(M, (2, 0), sym=range(2)); Sym2M
+            sage: Sym2M = M.tensor_module(2, 0, sym=range(2)); Sym2M
             Free module of fully symmetric type-(2,0) tensors
              on the Rank-3 free module M over the Integer Ring
 
         """
-        prefix, suffix = self._basis_comp()._repr_symmetry()
+        prefix, suffix = self._basis_sym()._repr_symmetry()
         return "Free module of {}type-({},{}) tensors on the {}{}".format(
             prefix.lower(), self._tensor_type[0], self._tensor_type[1], self._fmodule, suffix)
 
@@ -226,11 +222,10 @@ def _is_symmetry_coarsening_of(self, coarser_comp, finer_comp):
 
         EXAMPLES::
 
-            sage: from sage.tensor.modules.tensor_free_submodule import TensorFreeSubmodule_comp
             sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
             sage: e = M.basis('e')
             sage: T60M = M.tensor_module(6, 0)
-            sage: Sym0123x45M = TensorFreeSubmodule_comp(M, (6, 0), sym=((0, 1, 2, 3), (4, 5)))
+            sage: Sym0123x45M = M.tensor_module(6, 0, sym=((0, 1, 2, 3), (4, 5)))
             sage: ten0123x45M = Sym0123x45M.an_element(); ten0123x45M
             Type-(6,0) tensor on the Rank-3 free module M over the Integer Ring
             sage: ten0123x45M.parent()
@@ -242,13 +237,13 @@ def _is_symmetry_coarsening_of(self, coarser_comp, finer_comp):
              on the Rank-3 free module M over the Integer Ring,
              with symmetry on the index positions (0, 1, 2, 3),
              with symmetry on the index positions (4, 5)
-            sage: Sym012x345M = TensorFreeSubmodule_comp(M, (6, 0), sym=((0, 1, 2), (3, 4, 5)))
+            sage: Sym012x345M = M.tensor_module(6, 0, sym=((0, 1, 2), (3, 4, 5)))
             sage: com012x345M = Sym012x345M.an_element()._components[e]; com012x345M
             6-indices components w.r.t. Basis (e_0,e_1,e_2)
              on the Rank-3 free module M over the Integer Ring,
              with symmetry on the index positions (0, 1, 2),
              with symmetry on the index positions (3, 4, 5)
-            sage: Sym012345M  = TensorFreeSubmodule_comp(M, (6, 0), sym=((0, 1, 2, 3, 4, 5)))
+            sage: Sym012345M  = M.tensor_module(6, 0, sym=((0, 1, 2, 3, 4, 5)))
             sage: com012345M  = Sym012345M.an_element()._components[e]; com012345M
             Fully symmetric 6-indices components w.r.t. Basis (e_0,e_1,e_2)
              on the Rank-3 free module M over the Integer Ring
@@ -302,11 +297,10 @@ def _element_constructor_(self, comp=[], basis=None, name=None,
         r"""
         TESTS::
 
-            sage: from sage.tensor.modules.tensor_free_submodule import TensorFreeSubmodule_comp
             sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
             sage: e = M.basis('e')
             sage: T60M = M.tensor_module(6, 0)
-            sage: Sym0123x45M = TensorFreeSubmodule_comp(M, (6, 0), sym=((0, 1, 2, 3), (4, 5)))
+            sage: Sym0123x45M = M.tensor_module(6, 0, sym=((0, 1, 2, 3), (4, 5)))
             sage: Sym0123x45M(e[0]*e[0]*e[0]*e[0]*e[1]*e[2])
             Traceback (most recent call last):
             ...
@@ -322,14 +316,14 @@ def _element_constructor_(self, comp=[], basis=None, name=None,
         if sym is not None or antisym is not None:
             # Refuse to create a tensor with finer symmetries
             # than those defining the subspace
-            if not self._is_symmetry_coarsening_of((sym, antisym), self._basis_comp()):
+            if not self._is_symmetry_coarsening_of((sym, antisym), self._basis_sym()):
                 raise ValueError(f"cannot create a tensor with symmetries {sym=}, {antisym=} "
                                  f"as an element of {self}")
 
         if sym is None:
-            sym = self._basis_comp()._sym
+            sym = self._basis_sym()._sym
         if antisym is None:
-            antisym = self._basis_comp()._antisym
+            antisym = self._basis_sym()._antisym
 
         resu = super()._element_constructor_(comp=comp,
                                              basis=basis, name=name,
@@ -355,12 +349,11 @@ def is_submodule(self, other):
 
         EXAMPLES::
 
-            sage: from sage.tensor.modules.tensor_free_submodule import TensorFreeSubmodule_comp
             sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
             sage: T60M = M.tensor_module(6, 0)
-            sage: Sym0123x45M = TensorFreeSubmodule_comp(M, (6, 0), sym=((0, 1, 2, 3), (4, 5)))
-            sage: Sym012x345M = TensorFreeSubmodule_comp(M, (6, 0), sym=((0, 1, 2), (3, 4, 5)))
-            sage: Sym012345M  = TensorFreeSubmodule_comp(M, (6, 0), sym=((0, 1, 2, 3, 4, 5)))
+            sage: Sym0123x45M = M.tensor_module(6, 0, sym=((0, 1, 2, 3), (4, 5)))
+            sage: Sym012x345M = M.tensor_module(6, 0, sym=((0, 1, 2), (3, 4, 5)))
+            sage: Sym012345M  = M.tensor_module(6, 0, sym=((0, 1, 2, 3, 4, 5)))
             sage: Sym012345M.is_submodule(Sym012345M)
             True
             sage: Sym012345M.is_submodule(Sym0123x45M)
@@ -387,8 +380,8 @@ def is_submodule(self, other):
         if self_tensor_type != other_tensor_type:
             return False
 
-        other_comp = other._basis_comp()
-        return self._is_symmetry_coarsening_of(self._basis_comp(), other_comp)
+        other_comp = other._basis_sym()
+        return self._is_symmetry_coarsening_of(self._basis_sym(), other_comp)
 
     @lazy_attribute
     def lift(self):
@@ -397,9 +390,8 @@ def lift(self):
 
         EXAMPLES::
 
-            sage: from sage.tensor.modules.tensor_free_submodule import TensorFreeSubmodule_comp
             sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
-            sage: Sym0123x45M = TensorFreeSubmodule_comp(M, (6, 0), sym=((0, 1, 2, 3), (4, 5)))
+            sage: Sym0123x45M = M.tensor_module(6, 0, sym=((0, 1, 2, 3), (4, 5)))
             sage: Sym0123x45M.lift
             Generic morphism:
               From: Free module of type-(6,0) tensors on the Rank-3 free module M over the Integer Ring,
@@ -419,7 +411,6 @@ def reduce(self):
 
         EXAMPLES::
 
-            sage: from sage.tensor.modules.tensor_free_submodule import TensorFreeSubmodule_comp
             sage: M = FiniteRankFreeModule(QQ, 3, name='M')
             sage: e = M.basis('e')
             sage: X = M.tensor_module(6, 0)
@@ -461,8 +452,8 @@ def reduce(self):
             sage: all(Y.reduce(u.lift()) == 0 for u in Y.basis('e'))
             True
         """
-        sym = self._basis_comp()._sym
-        antisym = self._basis_comp()._antisym
+        sym = self._basis_sym()._sym
+        antisym = self._basis_sym()._antisym
 
         def _reduce_element(x):
             if not x._components:
@@ -491,7 +482,6 @@ def retract(self):
 
         EXAMPLES::
 
-            sage: from sage.tensor.modules.tensor_free_submodule import TensorFreeSubmodule_comp
             sage: M = FiniteRankFreeModule(ZZ, 3, name='M')
             sage: e = M.basis('e')
             sage: X = M.tensor_module(6, 0)
diff --git a/src/sage/tensor/modules/tensor_free_submodule_basis.py b/src/sage/tensor/modules/tensor_free_submodule_basis.py
index ae00f98971b..d2b60298ee0 100644
--- a/src/sage/tensor/modules/tensor_free_submodule_basis.py
+++ b/src/sage/tensor/modules/tensor_free_submodule_basis.py
@@ -1,5 +1,5 @@
 r"""
-Free module bases indexed by component indices
+Standard bases of free submodules of tensor modules defined by some monoterm symmetries
 """
 
 #******************************************************************************
@@ -14,9 +14,9 @@
 from sage.tensor.modules.free_module_basis import Basis_abstract
 
 
-class TensorFreeSubmoduleBasis_comp(Basis_abstract):
+class TensorFreeSubmoduleBasis_sym(Basis_abstract):
     r"""
-    Standard basis of a tensor module with prescribed symmetries.
+    Standard basis of a free submodule of a tensor module with prescribed monoterm symmetries.
 
     EXAMPLES::
 
@@ -52,7 +52,7 @@ def __init__(self, tensor_module, symbol, latex_symbol=None, indices=None,
                                               latex_indices, symbol_dual, latex_symbol_dual)
         super().__init__(tensor_module, symbol, latex_symbol, indices, latex_indices)
         self._base_module_basis = base_module_basis
-        self._comp = tensor_module._basis_comp()
+        self._comp = tensor_module._basis_sym()
 
     def _repr_(self):
         r"""
@@ -120,8 +120,8 @@ def __getitem__(self, index):
             sage: e11[1, 2].display()
             e_1⊗e^2
 
-            sage: from sage.tensor.modules.tensor_free_submodule import TensorFreeSubmodule_comp
-            sage: Sym2M = TensorFreeSubmodule_comp(M, (2, 0), sym=range(2)); Sym2M
+            sage: from sage.tensor.modules.tensor_free_submodule import TensorFreeSubmodule_sym
+            sage: Sym2M = TensorFreeSubmodule_sym(M, (2, 0), sym=range(2)); Sym2M
             Free module of fully symmetric type-(2,0) tensors on the Rank-3 free module M over the Integer Ring
             sage: eSym2M = Sym2M.basis('e')
             sage: eSym2M[1, 1].display()