Ideal magnetohydrodynamic (MHD) theory is used to investigate some of the fundamental properties of the geodesic acoustic continuum modes (GAMs) in tokamaks, including their global structure, their associated magnetic components both inside and outside the plasma, and effects of a non-circular cross section of the plasma. In addition to the well-known $m=1$ side-bands in the perturbed density and pressure of the (electrostatic) GAM, the MHD continuum GAM also includes a $m=1$ side-band in the perturbed toroidal magnetic field as well as $m=2$ side-bands in the perturbed density, pressure, poloidal flow and in the magnetic components $delta {{B}{r}}$ and $delta {{B}{ heta}}$ ($m$ is the poloidal mode number). These $m=2$ side-bands exist within the whole plasma and the magnetic components also outside the plasma, and the magnitudes of these components in the vacuum region are calculated in the paper. It is shown that, for plasmas with a conducting wall not too far from the plasma surface, the perturbed magnetic field in the vacuum region is dominated by its poloidal component $delta {{B}_{ heta}}$ , with poloidal dependence $sin 2 heta $ , in agreement with experiments. Aspects of the plasma equilibrium that affect the magnitude of the perturbed magnetic field in the vacuum region are discussed in the paper. Furthermore, the influence of a non-circular plasma cross section on the GAM frequency and on the spectrum of the global, perturbed magnetic field is analysed. It is found that the only significant effect of a non-circular cross section on the GAM frequency comes from elongation and its variation across the plasma radius. However, higher-order shaping effects, as well as finite aspect ratio, induce other Fourier components than $m=2$ in the magnetic halo that surrounds the GAM surface.