We characterize the identified sets of a wide range of stochastic choice models, including random utility, various models of boundedly-rational behavior, and dynamic discrete choice. In each of these settings, we show two distributions over choice rules are observationally equivalent if and only if they can be obtained from one another via a finite sequence of simple swapping transforms. We leverage this to obtain complete descriptions of both the defining inequalities and extreme points of these identified sets. In cases where choice frequencies vary smoothly with some parameters, we provide a novel global-inverse result for practically testing identification.