A ferroic alignment of a time-reversal-even axial vector, which is called the ferroaxial ordered state, is identified by the spontaneous rotational distortion of the lattice structure to break vertical mirror symmetry. We theoretically investigate what happens in the electronic structure once ferroaxial ordering occurs by analyzing a tetragonal $d\text{−}p$ cluster model with transition metal oxide in mind. By adopting the symmetry-adapted multipole representation, we elucidate the emergent cross-product form of the spin-orbit coupling by the interplay between the atomic spin-orbit coupling and the $d\text{−}p$ hybridization according to vertical mirror symmetry breaking, which corresponds to the electronic order parameter of ferroaxial ordering.

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• Multipole order
• Spin-orbit coupling
• Cluster models