More from the hip:
A magnet in translational motion with a velocity V will have an electric
dipole polarization P=(1/c)VxM . EG., if V is in the x direction and M is
in the y direction, P will be in the z direction - this is tantamount to a
polarized dielectric with two bound surface charge layers - positive on its
upper face and negative on its lower face (planes of constant z). This
electrically polarized slab has no electric monopole moment (it is overall
neutral) and no monopole E field. But it does have a dipole moment (P times
its volume), a vector in the z direction; and it will exhibit an external
dipole E field. (This much is NOT from the hip - this is "conventional
wisdom".)
Now (from the hip) spin a doughnut shaped disc magnet about its own axis.
Extrapolating the above, I would expect an electrical polarization to exist,
with bound cylindrical surface charge layers on its inner (say negative) and
outer (positive) curved surfaces. Now this electrically polarized slab has
zero monopole moment and ZERO DIPOLE MOMENT. It therefore produces zero
external monopole field and zero external dipole field. (In fact, it would
seem that even all higher order fields are also zero, like a charged
cylindrical capacitor).