Primordial magnetic fields and massive neutrinos can leave an interesting signal in the CMB temperature and polarization. We perform a systematic analysis of general perturbations in the radiation-dominated universe, accounting for any primordial magnetic field and including leading-order effects of the neutrino mass. We show that massive neutrinos qualitatively change the large-scale perturbations sourced by magnetic fields, but that the effect is much smaller than previously claimed. We calculate the CMB power spectra sourced by inhomogeneous primordial magnetic fields, from before and after neutrino decoupling, including scalar, vector and tensor modes, and consistently modeling the correlation between the density and anisotropic stress sources. In an appendix we present general series solutions for the possible regular primordial perturbations.