  Magnetic anisotropy is one of the important factors in determining magnetic
structures. A type of magnetic anisotropy is closely related to the symmetry of
crystals. We theoretically investigate magnetic anisotropy and its related
magnetic instability arising from an electric axial moment, which appears under
the breaking of the mirror symmetry parallel to the moment but does not require
the breakings of both spatial inversion and time-reversal symmetries. By
performing perturbation and mean-field calculations in a complementary way, we
show the appearance of the in-plane magnetic anisotropy when the electric axial
moment occurs, which tends to tilt in-plane spin moments from the crystal axes
in collaboration with relativistic spin--orbit coupling. We demonstrate such a
tendency for single-site and four-site cluster models, the latter of which
leads to the instability toward a spin vortex phase accompanying magnetic
monopole and magnetic toroidal dipole.
