  We study sorting operators $\mathbf{A}$ on permutations that are obtained
composing Knuth's stack sorting operator $\mathbf{S}$ and the reversal operator
$\mathbf{R}$, as many times as desired. For any such operator $\mathbf{A}$, we
provide a size-preserving bijection between the set of permutations sorted by
$\mathbf{S} \circ \mathbf{A}$ and the set of those sorted by $\mathbf{S} \circ
\mathbf{R} \circ \mathbf{A}$, proving that these sets are enumerated by the
same sequence, but also that many classical permutation statistics are
equidistributed across these two sets. The description of this family of
bijections is based on a bijection between the set of permutations avoiding the
pattern $231$ and the set of those avoiding $132$ which preserves many
permutation statistics. We also present other properties of this bijection, in
particular for finding pairs of Wilf-equivalent permutation classes.
