  A group $\Gamma$ with a family of subgroups $\mathbb{P}$ is relatively
hyperbolic if $\Gamma$ admits a cusp-uniform action on a proper
$\delta$--hyperbolic space. We show that any two such spaces for a given group
pair are quasi-isometric, provided the spaces have "constant horospherical
distortion," a condition satisfied by Groves--Manning's cusped Cayley graph and
by all negatively curved symmetric spaces. Consequently the Bowditch boundary
admits a canonical quasisymmetric structure, which coincides with the
"naturally occurring" quasisymmetric structure of the symmetric space when
considering lattices in rank one symmetric spaces.
  We show that a group $\Gamma$ is a lattice in a negatively curved symmetric
space $X$ if and only if a cusped space for $\Gamma$ is quasi-isometric to the
symmetric space. We also prove an ideal triangle characterization of the
$\delta$--hyperbolic spaces with uniformly perfect boundary due to Meyer and
Bourdon--Kleiner. An appendix concerns the equivalence of several definitions
of conical limit point found in the literature.
