Gibbs free energy, the fundamental thermodynamic potential used to calculate equilibrium mineral assemblages in geological systems, does not apply to non-hydrostatically stressed solids. Consequently, there is debate over the significance of non-hydrostatic stress in petrological and geophysical processes. To help resolve this debate, we consider the effects of non-hydrostatic stress on the polymorph pairs kyanite/sillimanite, graphite/diamond, calcite/aragonite, and quartz/coesite. While these polymorphs are most relevant to metamorphic processes, the concepts developed are applicable to any single-component solid reaction. We quantitatively show how stress variations normal to an interface alter equilibrium temperatures of polymorph pairs by approximately two orders of magnitude more than stress variations parallel to an interface. Thus, normal stress controls polymorph stability to first order. High-pressure polymorphs are expected to preferentially nucleate normal to and grow parallel to the maximum stress and low-pressure polymorphs, the minimum stress. Nonetheless, stress variations parallel to an interface allow for the surprising possibility that a high-pressure polymorph can become more stable relative to a low-pressure polymorph as stress decreases. The effects of non-hydrostatic stress on mineral equilibrium are unlikely to be observed in systems with interconnected, fluid-filled porosity, as fluid-mediated reactions yield mineral assemblages at approximately constant pressures. In dry systems, however, reactions can occur directly between elastic solids, facilitating the direct application of non-hydrostatic thermodynamics. Non-hydrostatic stress is likely to be important to the evolution of metamorphic systems, as preferential orientations of polymorphic reactions can generate seismicity and may influence fundamental rock properties such as porosity and seismic anisotropy.