Catalyzes the transport of 2-oxoglutarate (alpha-oxoglutarate) across the inner mitochondrial membrane in an electroneutral exchange for malate. Can also exchange 2-oxoglutarate for other dicarboxylic acids such as malonate, succinate, maleate and oxaloacetate, although with lower affinity. Contributes to several metabolic processes, including the malate-aspartate shuttle, the oxoglutarate/isocitrate shuttle, in gluconeogenesis from lactate, and in nitrogen metabolism.