co.combinatorics - Name of a particular conjugate permutation

This is the reverse-complement of $pi$.

In one-line notation, the reverse of a permutation is what you get by writing it backwards and the complement of a permutation is what you get when you replace each entry $i$ by $n -i + 1$. (In other words, one of these operations is multiplication by $chi$ on the right, the other on the left.) The reverse-complement is what you get by doing both of these operations, or equivalently by giving the permutation matrix a half-turn. (Together with inversion, these operations generate the dihedral group acting on each permutation matrix.)