Inversion symmetry forbids bulk second-order spin photocurrents driven solely by electric-dipole coupling. We show that interference between the electric-dipole and spin Zeeman couplings of light enables Zeeman spin-shift and spin-injection currents in centrosymmetric magnets. Our microscopic theory and systematic symmetry analysis establish the allowed responses and polarization selection rules for both spin-degenerate and spin-split antiferromagnets, with and without spin–orbit coupling (SOC). In collinear magnets without SOC, spin injection vanishes, whereas a pure spin-shift current survives independently of the carrier scattering time. This response arises from transverse electronic spin coherences generated by the optical magnetic field, while the ordered moments remain collinear and fixed. Model analyses validate the mechanism, and first-principles calculations for MnF₂ predict experimentally accessible spin photocurrents. These findings establish a general framework for optical spin-current generation in centrosymmetric magnets, including a SOC-free pure-spin mechanism particularly relevant to altermagnets.