  We consider nonholonomic Chaplygin systems and associate to them a $(1,2)$
tensor field on the shape space, that we term the gyroscopic tensor, and that
measures the interplay between the non-integrability of the constraint
distribution and the kinetic energy metric. We show how this tensor may be
naturally used to derive an almost symplectic description of the reduced
dynamics. Moreover, we express sufficient conditions for measure preservation
and Hamiltonisation via Chaplygin's reducing multiplier method in terms of the
properties of this tensor. The theory is used to give a new proof of the
remarkable Hamiltonisation of the multi-dimensional Veselova system obtained by
Fedorov and Jovanovi\'c in J. Nonlinear Sci. 2004, and Reg. Chaot. Dyn. 2009.
