  In this paper, we consider the hamiltonian formulation of nonholonomic
systems with symmetries and study several aspects of the geometry of their
reduced almost Poisson brackets, including the integrability of their
characteristic distributions. Our starting point is establishing global
formulas for the nonholonomic Jacobiators, before and after reduction, which
are used to clarify the relationship between reduced nonholonomic brackets and
twisted Poisson structures. For certain types of symmetries (generalizing the
Chaplygin case), we obtain genuine Poisson structures on the reduced spaces and
analyze situations in which the reduced nonholonomic brackets arise by applying
a gauge transformation to these Poisson structures. We illustrate our results
with mechanical examples, and in particular show how to recover several
well-known facts in the special case of Chaplygin symmetries.
