Let $g \in A_{n}$ be a permutation whose disjoint cycle decomposition consists of odd cycles of distinct lengths, say $m$ cycles with distinct odd lengths $r_1,\cdots,r_m$.
Prove that $g$ is conjugate to $g^{-1}$ if and only if $\sum_{j=1}^{m}\frac{r_j-1}{2}$ is even.
Remark: By the spliting criterion we know the class $g^{S_n}$ splits into two equal sized classes of $g$ in $A_n$ with representatives $g$ and $(1,2)^{-1}g(1,2)$ respectively. We need to restrict $r_1,\cdots,r_m$ which forces $g^{-1}$ lies in the class of $g$.