We study the symmetries of the post-recombination cosmological magnetohydrodynamical equations which describe the evolution of dark matter, baryons and magnetic fields in a self-consistent way. This is done both at the level of fluid equations and of Vlasov-Poisson-Maxwell equations in phase space. We discuss some consistency relations for the soft limit of the (n+1)-correlator functions involving magnetic fields and matter overdensities. In particular, we stress that any violation of such consistency relations at equal-time would point towards an inflationary origin of the magnetic field.