Fundamental particles come with magnetic and electric charge, which makes the particles into a monopole source for the magnetic and electric fields. Of course, the magnetic charge is zero for all particles we know, because magnetic monopoles haven't been demonstrated to exist.
Fundamental particles also come with magnetic and electric spin, which makes the particles into a dipole source for the magnetic and electric fields. Here most particles have a nonzero magnetic dipole moment (the spin), but a zero or very small electric dipole moment.
Is there anything analogous to higher-order moments? Is there like an analog of the spin of a particle but for higher moments? If not, is there a reason why this isn't plausible to consider?