Justify geometrically: an element and its inverse are not conjugate

Question

Consider the group $G$ of rotations of regular tetrahedron in $\mathbb{R}^3$. We know that this group is $A_4$. We also know that a rotation of order $3$ and its inverse are not conjugate: ratation of order $3$ corresponds to a 3-cycle $(123)$ and in $A_4$ we know by algebraic arguments that $(123)$ and $(132)$ are not conjugate.

Q. Is there any geometric smart way to show that a rotation of order $3$ and its inverse are not conjugate in the group of rotational symmetries?

Answer 1

You can visualize this as follows: The tetrahedron is orientable, that is you can draw on each face a circular arc such that where these orientations meet on the edges they annihilate each other. There are six rotations, two for each face : a positive one and a negative one. It is impossible for the action of the symmetry group to map a positive rotation into a negative one.