= Tetrahedral rotation conjugacy classes
{title2=$1,\quad3,\quad4,\quad4$}
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 $S_4$.
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