Electric and magnetic parts in Maxwell's equations are kind of similar, so physical effects relating these properties have sometimes 'dual' analogues - with exchanged places.

For example in Aharonov-Bohm effect, the phase of charged particle depends on side of magnetic flux tube it comes through, while in its 'dual' analogue: Aharonov-Casher, the particle has magnetic moment and tube contains line of charge (it was used e.g. for neutron interference).

Another interesting 'dual' effect (hypothetical) can be found in magnetic monopole Wikipedia article - full expression for Lorenz force in such case would be:

where is magnetic charge - the last term corresponds to magnetic monopole - electric field interaction.

The question is if we should expect similar term for not only magnetic monopoles, but also for much more common: magnetic moments(dipoles)?

I would say that yes - for example imagine classical electron traveling in proton's electric field - let's change reference frame such that for infinitesimal time electron stops and proton is moving in also magnetic field created by quite large electron's magnetic moment - because of 3rd Newton's law, resulting Lorentz force should also work on electron ...

(3) equation here is Lagrangian for such electron's movement:

where the last term would correspond to such eventual magnetic moment-electric field interaction.

I wanted to ask if such looking quite important force is true or not?

If true, it seems to be completely forgotten - there would be needed some better sources ... have you seen something like that in a book or paper?

ps. Quantum mechanics was built on classical one, so there should be some correspondence in QM to the last term in Lagrangian above (making Bohr's orbits unstable) - atomic spin-orbit interaction seems to be one of them (?)