We introduce a generalization of the set of completely positive matrices called "pairwise completely positive" (PCP). These are pairs of matrices that share a joint decomposition so that one of them is necessarily positive semidefinite, while the other one is necessarily entrywise non-negative. We explore basic properties of these matrix pairs, present several testable necessary and sufficient conditions for a pair to be PCP, and we explore how this PCP property is connected to entanglement of quantum states.