  We consider the involutions known as "toggles," which have been used to give
simplified proofs of the fundamental properties of the promotion and evacuation
maps. We transfer these involutions so that they generate a group $\mathscr
P_n$ that acts on the set $S_n$ of permutations of $\{1,\ldots,n\}$. After
characterizing its orbits in terms of permutation skeletons, we apply the
action in order to understand West's stack-sorting map. We obtain a very simple
proof of a result that clarifies and extensively generalizes a theorem of
Bouvel and Guibert and also generalizes a theorem of Bousquet-M\'elou. We also
settle a conjecture of Bouvel and Guibert. We prove a result related to the
recently-introduced notion of postorder Wilf equivalence. Finally, we
investigate an interesting connection among the action of $\mathscr P_n$ on
$S_n$, the group structure of $S_n$, and the stack-sorting map.
