I've been working on sorting and factorisation problems on permutations for some time now, and have observed that given a permutation $\pi$ of $n$ elements, the permutation $\pi^\chi=\chi\circ\pi\circ\chi^{-1}$, where $\chi=\chi^{-1}=(n\ n-1\ \cdots\ 2\ 1)$, often has attractive properties (with respect to a particular sorting problem).

Is there a name for this "special" permutation (other than "the conjugate of $\pi$ by $\chi$")?