This article presents a comprehensive characterization of the thermoluminescence (TL) and optically stimulated luminescence (OSL) relative luminescence efficiencies of carbon-doped aluminum (Al2O3:C) for heavy charged particles (HCPs) with atomic numbers ranging from 1 (proton) to 54 (xenon) and energies ranging from 7 to 1000 MeV/u, and investigates the dependence of the Al2O3:C response on experimental conditions. Relative luminescence efficiency values are presented for 19 primary charge/energy combinations, plus 31 additional charge/energy combinations obtained by introducing absorbers in the primary beam. Our results show that for energies of hundreds of MeV/u the data can be described by a single curve of relative luminescence efficiency versus linear energy transfer (LET). This information is needed to compensate for the reduced OSL efficiency to high-LET particles in such applications as space dosimetry. For lower energies, the relative luminescence efficiency as function of LET cannot be described by a single curve; instead, it separates into different components corresponding to different particles. We also present data on the low-LET dose response of Al2O3:C, measured under the same experimental conditions in which the relative luminescence efficiencies to HCPs were obtained, providing information relevant to future theoretical investigations of HCP energy deposition and luminescence production in Al2O3:C.