Blue is "normal" magnetism; Red is co-magnetism. Normal magnetism spreads out as lines repel and produces the conventional, toroidal magnetic field. Co-magnetism bunches together in the middle and shoots out of the poles. (The flat top and bottom of the red lines are an artifact of "clipping", since the function would otherwise go to infinity.)
This depiction is when there are 2 units of magnetic ionization present in a single, scalar dimension. I figured that if thermal ionization can go into a second (intermediate) and third (ultra-high) unit, both electric and magnetic ionization should also be able to.
In co-magnetism, like attracts like, so in an aggregate, the atoms would end up with north poles together and south poles together, alternating. This may explain where antiferromagnetic ordering comes from.
Co-magnetism would attract along the verticals (like poles), and the horizontals would attract because each atom would be flipped to its adjacent, with opposite poles attracting. Therefore, the magnetic system would be stable.