  We study locally compact metric spaces that enjoy various forms of
homogeneity with respect to M\"obius self-homeomorphisms. We investigate
connections between such homogeneity and the combination of isometric
homogeneity with invertibility. In particular, we provide a new
characterization of snowflakes of boundaries of rank-one symmetric spaces of
non-compact type among locally compact and connected metric spaces.
Furthermore, we investigate the metric implications of homogeneity with respect
to uniformly strongly quasi-M\"obius self-homeomorphisms, connecting such
homogeneity with the combination of uniform bi-Lipschitz homogeneity and
quasi-invertibility. In this context we characterize spaces containing a cut
point and provide several metric properties of spaces containing no cut points.
These results are motivated by a desire to characterize the snowflakes of
boundaries of rank-one symmetric spaces up to bi-Lipschitz equivalence.
