A group is said to be ambivalent all its elements are conjugate to their inverse. For example, every symmetric group is ambivalent, as two elements of $S_n$ are conjugate iff they share the same cycle structure, which of course is the case for an element and its inverse. (Note we don't require conjugating by the same element in each case.)
I'm interested in the case of alternating groups. It's not difficult to show that $A_n$ is ambivalent only for $n \in \{1,2,5,6,10,14\}$; one mostly just needs to understand conjugacy classes in the alternating group. I'm not asking about this proof, but rather, if there are other more geometric (or otherwise revealing) ways to understand particular cases.
For example, $A_5$ is isomorphic to the rotational symmetry group of an icosahedron, and geometrically it is clear that that group is ambivalent (at least, if one has a feel for conjugacy). I'm curious if for other larger $n$ there are interesting ways to understand/visualize/intuit this, whether geometric or otherwise.
