Geometrical meaning of A4 conjugacy classes of elements of order 3 I know that elements of order 3 in A4 group split into two separate conjugacy classes, due to lack of odd elements to form a unique class like happens in the full S4.
Since A4 is the rotation symmetry group of the Tetrahedron, I was wondering if these two split conjugacy classes have a direct geometrical meaning. While I can see a direct meaning of transpositions, clearly distinguishing them from 3-cycles, I cannot see any apparent related split of them from a geometrical perspective.
Nevertheless, my feeling is this binding makes sense, hiding somewhere from my view.
Would you point me to the right direction, please?
Thanks in advance
 A: For me, it is helpful to see the conjugacy classes of $A_4$ listed out:

Now, let there be a tetrahedron sitting on a desk. Label the three vertices on the desk $1,2,3$, going clockwise, and label the vertex at the top as $4$. Now, if you rotate the tetrahedron $120^\circ$ degrees clockwise, that would be like the cycle $(123)$. Rotate it counterclockwise, however, and that would be like the cycle $(132)$.
Now, turn the tetrahedron so that the vertex $1$ is now on top. You will now notice that, in clockwise order, the vertices on the desk are $2,4,3$. Thus, rotate the tetrahedron $120^\circ$ degrees clockwise, and that would be like the cycle $(243)$. Rotate it counterclockwise, and that would be like the cycle $(234)$.
Again, turn the tetrahedron so that the vertex $2$ is now on top. In  clockwise order, the vertices on the desk are now $1,3,4$. Thus, rotate the tetrahedron $120^\circ$ degrees clockwise, and that would be like the cycle $(134)$. Rotate it counterclockwise, and that would be like the cycle $(143)$.
Hopefully, you can guess what will happen if you turn the tetrahedron so that the vertex $3$ is on top. Anyway, I hope you can now see how the conjugacy class with $(123)$ corresponds to clockwise rotations when looking from above while the conjugacy class with $(132)$ corresponds to counter-clockwise rotations when looking from above.
