Dzyaloshinskii-Moriya interaction (DMI) at metallic multilayer interfaces of Co thin films and heavy-metals X (X=Ir, Pt) was investigated from first principles calculations that treat spin-spirals with the spin-orbit coupling. The results predict that the sign of the DMI parameters depends on the heavy-metals at the interfaces, and that the variation of the orbital moments behaves differently depending on the sign of the DMI parameters.