# how to check if a transform contains a mirror reflection?

On a 2D plane, a transform containing a mirror reflection will change a directed acyclic graph from clockwise to anti-clockwise. In math this can be done that given a triangle ABC, we can check which direction the cross product $$\vec{AB} \times \vec{BC}$$ is pointing to, e.g. if the plane ABC is on $$x-y$$ space, sign of $$\vec{AB} \times \vec{BC}$$ on $$z$$ direction can serve as the criterion.

How about in the 3D space? Given a transform $$\tau$$ which transformed cube $$ABCDEFGH$$ into $$A'B'C'D'E'F'G'H'$$, or a regular tetrahedron $$ABCD$$ into $$A'B'C'D'$$, is there a way to check if it's gone through a mirror reflection?

How is the case in 4D or even higher space?