<p>The purpose of this paper is twofold. The first purpose is to prove that there is a one-to-one correspondence between the rough quasi-isometry classes of proper, geodesic, roughly starlike and Gromov hyperbolic spaces and the relatively power quasi-symmetry with RBI classes of bounded uniform metric spaces. This is a counterpart of the main result of Bonk, Heinonen, and Koskela’s paper in 2001 in the rough quasi-mapping setting. As an application, we prove that two bounded uniform metric spaces are roughly quasi-isometrically equivalent with respect to the quasi-hyperbolic metrics if and only if the metric boundaries of these two spaces are power quasi-symmetrically equivalent. The second purpose is to characterize the quasi-isometry of a homeomorphism in the quasi-hyperbolic metrics in terms of the local quasi-isometry of this homeomorphism and its inverse, which is a continuation of the related study started by Bonk, Heinonen, and Koskela in 2001.</p>