@SamHopkins I wanted to point out that, at least for Weyl groups, this is not just a Coxeter element phenomenon. Every element $g\in W$ is conjugate to its inverse via an involution! This is a main result of Carter's "Conjugacy classes in the Weyl group", see Thm. C: (iii). So maybe you can call it something like "inverting involution" or "Carter's involution" etc. Carter's proof does involve special bipartite diagrams for all conjugacy classes, but you don't have something as nice as a Coxeter plane. Also, importantly, his proof is case-by-case. I do not think there is a uniform one.