We present a new methodology for inverting P-to-S receiver function (RF) waveforms directly for mantle temperature and composition. This is achieved by interfacing the geophysical inversion with self-consistent mineral phase equilibria calculations from which rock mineralogy and its elastic properties are predicted as a function of pressure, temperature, and bulk composition. This approach anchors temperatures, composition, seismic properties, and discontinuities that are in mineral physics data, while permitting the simultaneous use of geophysical inverse methods to optimize models of seismic properties to match RF waveforms. Resultant estimates of transition zone (TZ) topography and volumetric seismic velocities are independent of tomographic models usually required for correcting for upper mantle structure. We considered two end-member compositional models: the equilibrated equilibrium assemblage (EA) and the disequilibrated mechanical mixture (MM) models. Thermal variations were found to influence arrival times of computed RF waveforms, whereas compositional variations affected amplitudes of waves converted at the TZ discontinuities. The robustness of the inversion strategy was tested by performing a set of synthetic inversions in which crustal structure was assumed both fixed and variable. These tests indicate that unaccounted-for crustal structure strongly affects the retrieval of mantle properties, calling for a two-step strategy presented herein to simultaneously recover both crustal and mantle parameters. As a proof of concept, the methodology is applied to data from two stations located in the Siberian and East European continental platforms.