  We define $\partial$-biLipschitz homeomorphisms between uniform metric spaces
and show that these maps are always quasim\"obius. We also show that a
homeomorphism being $\partial$-biLipschitz is equivalent to the map biLipschitz
in the quasihyperbolic metrics on these spaces. The proofs of these claims
require us to uniformize the quasihyperbolic metric. We further show that all
admissible uniformizations of a Gromov hyperbolic space are quasim\"obius to
one another by the identity map, including those uniformizations that are based
at a point of the Gromov boundary. Using the main results we then show that the
sphericalization and inversion operations are compatible with uniformization of
hyperbolic spaces in a natural sense.
