  A nonholonomic system is a mechanical system with velocity constraints not
originating from position constraints; rolling without slipping is the typical
example. A nonholonomic integrator is a numerical method specifically designed
for nonholonomic systems. It has been observed numerically that many
nonholonomic integrators exhibit excellent long-time behaviour when applied to
various test problems. The excellent performance is often attributed to some
underlying discrete version of the Lagrange--d'Alembert principle. Instead, in
this paper, we give evidence that reversibility is behind the observed
behaviour. Indeed, we show that many standard nonholonomic test problems have
the structure of being foliated over reversible integrable systems. As most
nonholonomic integrators preserve the foliation and the reversible structure,
near conservation of the first integrals is a consequence of reversible KAM
theory. Therefore, to fully evaluate nonholonomic integrators one has to
consider also non-reversible nonholonomic systems. To this end we construct
perturbed test problems that are integrable but no longer reversible (with
respect to the standard reversibility map). Applying various nonholonomic
integrators from the literature to these problems we observe that no method
performs well on all problems. This further indicates that reversibility is the
main mechanism behind near conservation of first integrals for nonholonomic
integrators. A list of relevant open problems is given.
