Experiments in small aspect-ratio Taylor–Couette flows have reported the presence of a band in parameter space where rotating waves become steady non-axisymmetric solutions (a pinning effect) via infinite-period bifurcations. Previous numerical simulations were unable to reproduce these observations. Recent additional experiments suggest that the pinning effect is not intrinsic to the dynamics of the problem, but rather is an extrinsic response induced by the presence of imperfections. Here we present numerical simulations that include a small tilt of one of the endwalls, simulating the effects of imperfections that break the SO(2) axisymmetry of the problem, and indeed are able to reproduce the experimentally observed pinning of the rotating waves. Dynamical systems considerations suggest that any imperfection breaking the SO(2) axisymmetry of the problem must result in the formation of a pinning region of finite width. We have also found that the particulars of the pinning process, in particular the width of the pinning region, are extremely sensitive to the type of imperfection in the system. Almost identical flows respond in completely different ways to the same imperfection, depending on subtle differences in the weak secondary characteristics of the flow. The numerical simulations of the Navier–Stokes equations for the problem with an imposed tilt of an endwall together with normal-form analysis of a Hopf bifurcation subjected to imposed symmetry-breaking help shed some light on previous experiments that reported a variety of different dynamical behaviour for which a clear explanation was lacking.