 Tetrahedral rotation conjugacy classes

The proper rotational tetrahedral symmetry group is $A_4$, acting by even permutations of the vertices. Its conjugacy classes consist of the identity, the three half-turns about opposite-edge midpoint axes, four rotations through $+2\pi /3$ about outward-pointing vertex axes, and their four inverses. A 3-cycle has a three-element centralizer, so its class has size $12/3=4$. Conjugating it to its inverse would require an odd permutation on its three moved vertices, so the two orientations form different classes in $A_4$ even though they lie in one class in $S_4$.