The internal dynamics of a dark matter structure may have the remarkable property that the local temperature in the structure depends on direction. This is parameterized by the velocity anisotropy β which must be zero for relaxed collisional structures, but has been shown to be nonzero in numerical simulations of dark matter structures. Here, we present a method for inferring the radial profile of the velocity anisotropy of the dark matter halo in a galaxy cluster from X-ray observables of the intracluster gas. This nonparametric method is based on a universal relation between the dark matter temperature and the gas temperature which is confirmed through numerical simulations. We apply this method to observational data and we find that β is significantly different from zero at intermediate radii. Thus, we find a strong indication that dark matter is effectively collisionless on the dynamical timescale of clusters, which implies an upper limit on the self-interaction cross-section per unit mass σ/m 1 cm2 g–1. Our results may provide an independent way to determine the stellar mass density in the central regions of a relaxed cluster, as well as a test of whether a cluster is in fact relaxed.