Follow up on when it happens that $b=b^{-1}$ This is a question that came up after my previous question here.
In my previous question I asked for help to prove that if $a$ is a group element of odd order and $b$ an element such that $b = a b^{-1}a^{-1}$ then $b^2 =e$.
Then I tried to understand when it happens that $b = b^{-1}$ and I tried to make a general statement. 
In this case we found an element of odd order so that $b$ and $b^{-1}$ are conjugate and then it followed that they are in fact equal.
But it seems plausible to me that whenever $b$ and $b^{-1}$ are conjugate then one should be able to say more about $b$ or $b^{-1}$.

Is there a theorem about what holds true when $b$ and $b^{-1}$ are
  conjugate? Or perhaps a similar statement along these lines?


This has to to with real conjugacy classes, and also ambivalent groups (A group is said to be ambivalent if every element is conjugate to its inverse). See for example here. Of course every $S_n$ is ambivalent. But not all $A_n$ are ambivalent.
