Self-similar solutions are presented for the equatorial propagation of axisymmetric, piston-driven magnetohydrodynamic shocks into an inhomogeneous ideal gas (∼ r-ω, 0≤ω<3) permeated by an azimuthal magnetic field (∼r-m, 0<m< 1.5). Several regimes of magnetically dominated flow near the piston are possible (depending on the ambient density and field distribution of the unshocked gas) along with a new quasi-hydrodynamic flow where the magnetic field is depressed near the piston. Here, the hydrodynamic pressure replaces the magnetic pressure as the means of transmitting forces to the gas adjacent to the piston. Since in each regime gradients of different quantities diverge, the results indicate that in each case different effects of dissipation and diffusion may thus be needed. The theory is applied to describe shock waves in a gravitational field due to a heavy nucleus at the origin.