What can we say about random variables such that it and its inverse have the same distribution? One example is Cauchy distributed random variables, easily proved via the fact that if $X, Y$ are IID standard normal then both $X/Y$ and $Y/X$ are Cauchy distributed.
If we restrict to random variables having a density, we can show that the density function $f$ must satisfy the functional equation $$ u(1/t)=t^2 u(t) $$ for $t\not =0$. This math SE post and this one have information on solutions.
I'm interested both in general solutions and solutions when restricted to symmetry, and in references.