This study reports a combined experimental and theoretical study of the solid-state polymorphism of the anticancer agent cisplatin. A complete assignment was performed for the inelastic neutron scattering (INS) and Raman spectra collected simultaneously for cisplatin, at different temperatures, with a view to obtain reliable and definitive evidence of the relative thermal stability of its alpha and beta polymorphic species. A marked temperature-dependent hysteresis was observed, as previously reported in the literature. Theoretical calculations were carried out at the density functional theory level, using a plane-wave basis set approach and pseudopotentials. A detailed comparison with the experimental Raman and INS data showed that the alpha polymorph is present at the lowest temperatures, whereas the beta form occurs near room temperature. Furthermore, regions of coexistence of both forms are identified, which depend on the working mode (heating or cooling). These findings imply that Raman spectroscopy allows clear identification of the alpha and beta polymorphs at a given temperature and can unambiguously discriminate between them. Elucidation of the polymorphic equilibrium of this widely used anticancer drug is paramount for its pharmaceutical preparation and storage conditions.