  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.
