Let $m$ be a positive integer. Suppose that $g$ and $h$ are two primitive roots modulo $m$. Denote by $\textrm{ind}_gh$ the index of $h$ to $g$ modulo $m$. Show that $(\textrm{ind}_gh)^2\equiv1(\textrm{mod }\varphi(m))$, where $\varphi$ is the Euler's function.
In fact, it is easy to show that $(\textrm{ind}_gh)(\textrm{ind}_hg)\equiv1(\textrm{mod }\varphi(m))$. However, I do not know how to show $\textrm{ind}_gh\equiv\mathrm{ind}_hg(\textrm{mod }\varphi(m))$. Further, I test $m=3$, $m=4$, $m=6$, and $m=18$, the modulo equations are all true. That is, I cannot prove it or give a counterexample. I shall be grateful if experts in Number Theory give me any hint. Thanks a lot!