There is a sense in which all "interesting" properties of functions in spherical geometry are invariant under conjugation by a Möbius transformation. The reason is that the Möbius transformations correspond to "uninteresting" manipulations of the whole sphere, as illustrated in this video.
Is there an equivalent notion in hyperbolic geometry? In other words, is there a valid statement of the form "All interesting properties of functions in hyperbolic geometry are invariant under conjugation by a $\text{Foo}$ because the $\text{Foo}$s correspond to uninteresting manipulations of the {upper half plane, unit disk, hyperboloid, etc}."?