diff --git a/docs/src/lib/operations.md b/docs/src/lib/operations.md
index d388ace301..b694e0fd00 100644
--- a/docs/src/lib/operations.md
+++ b/docs/src/lib/operations.md
@@ -109,6 +109,7 @@ isempty(::Intersection)
 ∈(::AbstractVector{Real}, ::Intersection{Real})
 isempty_known(::Intersection)
 set_isempty!(::Intersection, ::Bool)
+swap(::Intersection)
 use_precise_ρ
 _projection
 ```
diff --git a/src/Approximations/overapproximate.jl b/src/Approximations/overapproximate.jl
index d771bcf83c..f2befa89dc 100644
--- a/src/Approximations/overapproximate.jl
+++ b/src/Approximations/overapproximate.jl
@@ -305,5 +305,5 @@ function overapproximate(cap::Intersection{N,
                          dir::AbstractDirections{N};
                          kwargs...
                         ) where {N<:Real}
-    return overapproximate(cap.Y ∩ cap.X, dir; kwargs...)
+    return overapproximate(swap(cap), dir; kwargs...)
 end
diff --git a/src/Intersection.jl b/src/Intersection.jl
index 54a0a7582e..40b18bce50 100644
--- a/src/Intersection.jl
+++ b/src/Intersection.jl
@@ -3,6 +3,7 @@ import Base: isempty, ∈, ∩
 export Intersection,
        isempty_known,
        set_isempty!,
+       swap,
        use_precise_ρ,
        IntersectionArray,
        array
@@ -92,11 +93,14 @@ struct Intersection{N<:Real, S1<:LazySet{N}, S2<:LazySet{N}} <: LazySet{N}
     cache::IntersectionCache
 
     # default constructor with dimension check
-    function Intersection{N, S1, S2}(X::S1, Y::S2) where
+    function Intersection{N, S1, S2}(X::S1,
+                                     Y::S2;
+                                     cache::IntersectionCache=IntersectionCache()
+                                    ) where
             {N<:Real, S1<:LazySet{N}, S2<:LazySet{N}}
         @assert dim(X) == dim(Y) "sets in an intersection must have the same " *
             "dimension"
-        return new{N, S1, S2}(X, Y, IntersectionCache())
+        return new{N, S1, S2}(X, Y, cache)
     end
 end
 
@@ -150,6 +154,30 @@ function set_isempty!(cap::Intersection, isempty::Bool)
     return set_isempty!(cap.cache, isempty)
 end
 
+"""
+    swap(cap::Intersection{N, S1, S2})::Intersection{N} where {N<:Real, S1, S2}
+
+Return a new `Intersection` object with the arguments swapped.
+
+### Input
+
+- `cap` -- intersection of two convex sets
+
+### Output
+
+A new `Intersection` object with the arguments swapped.
+The old cache is shared between the old and new objects.
+
+### Notes
+
+The advantage of using this function instead of manually swapping the arguments
+is that the cache is shared.
+"""
+function swap(cap::Intersection{N, S1, S2}
+             )::Intersection{N} where {N<:Real, S1, S2}
+    return Intersection{N, S2, S1}(cap.Y, cap.X, cache=cap.cache)
+end
+
 
 # --- LazySet interface functions ---
 
@@ -351,7 +379,7 @@ function ρ(d::AbstractVector{N},
                              <:LazySet{N}};
            algorithm::String="line_search",
            kwargs...) where N<:Real
-    return ρ_helper(d, cap.Y ∩ cap.X, algorithm; kwargs...)
+    return ρ_helper(d, swap(cap), algorithm; kwargs...)
 end
 
 """
@@ -407,7 +435,7 @@ end
 function ρ(d::AbstractVector{N},
            cap::Intersection{N, <:AbstractPolytope{N}, <:LazySet{N}};
            kwargs...) where {N<:Real}
-    return ρ(d, cap.Y ∩ cap.X; kwargs...)
+    return ρ(d, swap(cap); kwargs...)
 end
 
 # disambiguation
@@ -436,7 +464,7 @@ function ρ(d::AbstractVector{N},
                              <:AbstractPolytope{N}};
            algorithm::String="line_search",
            kwargs...) where N<:Real
-    return ρ_helper(d, cap.Y ∩ cap.X, algorithm; kwargs...)
+    return ρ_helper(d, swap(cap), algorithm; kwargs...)
 end
 
 """