We proceed for the first time to the evaluation of the Born cross section for J/ψ+ηc production, namely, via g+g→J/ψ+ηc+g, and show that it has a harder PT spectrum than the J/ψ-pair yield at the Born level. If one stuck to a comparison at the Born level, one would conclude that J/ψ+ηc production would surpass that of J/ψ+J/ψ at large PT. This is nonetheless not the case since J/ψ-pair production, as for single J/ψ, receives leading-PT contributions at higher orders in αs. We also present the first evaluation of these leading-PT next-to-leading order contributions. These are indeed significant for increasing PT and are of essential relevance for comparison with forthcoming data. We also compute kinematic correlations relevant for double-parton-scattering studies. Finally, we evaluate the polarization of a J/ψ accompanied by either an ηc or a J/ψ and another light parton. These results may be of great help to understand the polarization of quarkonia produced at high energies.