Could two celestial bodies with extreme magnetic fields affect each other's motion via magnetism?

Assume we have two objects that exist on a planetary / stellar scale.

It could be a pair of magnetars, black holes, or really really big neodymium magnets. All that matters is the scale, and the magnetism.

What happens when the magnetic fields of these objects interfere? Is it possible to observe a change in motion that resists gravity? Do they tear each other apart?

And, most importantly, Have we observed an interaction like this?

The gravitational force declines as $$1/r^2$$ while magnetic field from a dipole goes as $$1/r^3$$: hence magnetic forces typically become negligible over longer distance, unless the source is extremely strong. But there are certainly extremely magnetic objects like magnetars, and for an orbiting object the direct gravitational force is replaced by gravitational tides that go as $$1/r^3$$, so sometimes this matters.
(Bromley and Kenyon 2022) looked at a small magnetic dipole object orbiting a heavy dipole, ignoring torques on it. They noted that if the dipole is oriented opposite to the big dipole there is an minimum stable circular orbit at $$r_{msco}=\sqrt{\frac{3\xi_r B_*R_*^3}{GM_sM_*}}$$ where $$R_*,M_*$$ is the radius and mass of the big dipole, $$B_*$$ the surface field at the equator, $$M_s$$ is the mass of the orbiting dipole and $$\xi_r$$ is the magnetic moment of the orbiter. This radius is small unless $$\xi_rB_*$$ are big compared to the gravitational force, so for most objects nothing happens. This actually includes most white dwarfs and neutron stars. But for the most extreme one the orbiter will spiral in (also, if it is a conductor with resistance, induced currents will dissipate orbital energy and make it spiral inward). If the orbiter has parallel magnetic poles there is repulsion instead, and it will orbit further out. In either case there is also orbital precession due to the magnetic effects, and this looks like the main realistic (and observable) effect.
What about more equal masses, and non-parallel fields? The force between two dipoles actually goes as $$1/r^5$$, (Yung, Landecker & Villani 1998), while the torque they exert on each other is $$1/r^3$$ (Landecker, Villani & Yung 1999): they are much more effective in turning each other's axis, causing spin precession. So it is unlikely the magnetic fields will have much effect on the orbits unless the masses are very close and very magnetic, basically requiring something like charged near extremal black holes (in which case relativistic effects make the issue even more confusing). But for strongly magnetic objects like neutron stars they might actually induce big torques on each other, causing the rotation axes to shift in complex ways. I think one needs to do numerical simulations here, but there is likely potential for all sorts of chaos and resonances (see below).