The magnetic spin-orbit contribution to nuclear Coulomb energies is enhanced by the nucleon's coupling to the strong relativistic nuclear scalar potential in the Dirac-Hartree-Fock theory. An enhancement factor of 40-60% is found in a number of light nuclei in the oscillator shell model. The effect reduces the Okamoto-Nolen-Schiffer anomaly in single-hole states, but has the wrong sign in single-particle states.