  Superexchange calculation is performed for multi-orbital band models with
broken inversion symmetry. Orbital-changing hopping terms allowed by the
symmetry breaking electric field lead to a new kind of orbital exchange term
closely resembling the Dzyaloshinskii-Moriya spin exchange. The final
superexchange Hamiltonian in two dimensions is expressed in terms of Gell-Mann
matrices that break down as three spin operators and five nematic (quadrupole)
order operators. Mean-field phase diagram exhibits a rich structure including
anti-ferro-orbital, ferro-orbital, and both single and multiple spiral-orbital
phases. Superexchange calculation for spinful, two-orbital model also strongly
suggests a robust chiral-orbital order in the ground state.
