[Math] Visualizing $180^\circ$ rotational symmetries of a tetrahedron

Question

I am trying to learn about the symmetries of a regular tetrahedron.

I understand the identity and all eight $120^\circ$ rotations that keep one vertex fixed, $(123),(132),(243),(234),(134),(143),(124),(142)$ but I cannot at all understand how to visualize the so-called $180^\circ$ rotations, i.e., $(13)(24)$ , $(14)(23)$ , $(12)(34)$.

Can anyone suggest anything for this?

Answer 1

Only that you can place a regular tetrahedron in a regular cube. Vertices, for example, at $$ (1,1,1); \; \; (1,-1,-1); \; \; (-1,1,-1); \; \; (-1,-1,1). $$ Each pair should disagree in two coordinates, agree in one.