In a thorough paper Kuchar has examined the canonical reduction of the most general action functional describing the geometrodynamics of the maximally extended Schwarzschild geometry. This reduction yields the true degrees of freedom for (vacuum) spherically symmetric general relativity (SSGR). The essential technical ingredient in Kuchar's analysis is a canonical transformation to a certain chart on the gravitational phase space which features the Schwarzschild mass parameter , expressed in terms of what are essentially Arnowitt - Deser - Misner variables, as a canonical coordinate. (Kuchar's paper complements earlier work by Kastrup and Thiemann, based mostly on Ashtekar variables, which has also explicitly isolated the true degrees of freedom for vacuum SSGR.) In this paper we discuss the geometric interpretation of Kuchar's canonical transformation in terms of the theory of quasilocal energy - momentum in general relativity given by Brown and York. We find Kuchar's transformation to be a `sphere-dependent boost to the rest frame', where the `rest frame' is defined by vanishing quasilocal momentum. Furthermore, our formalism is general enough to cover the case of (vacuum) two-dimensional dilaton gravity. Therefore, besides reviewing Kuchar's original work for Schwarzschild black holes from the framework of hyperbolic geometry, we present new results concerning the canonical reduction of Witten black-hole geometrodynamics. Finally, addressing a recent work of Louko and Whiting, we discuss some delicate points concerning the canonical reduction of the `thermodynamical action', which is of central importance in the path-integral formulation of gravitational thermodynamics.