Bubblesort, stacksort and their duals

Let BB and SS be, respectively, the base steps of bubblesort and stacksort  , and call B̃ and S̃ their dual versions via the reverse-complement map. We find some unexpected commutation properties between the classical operators and their duals, and we also prove that the set of permutations sortable by a prescribed number of iterations of BB and B̃ is a pattern class.