  I propose the emergence of the spin-orbital-coupled vector chirality in a
non-frustrated Mott insulator with the strong spin-orbit coupling due to
$ab$-plane's inversion-symmetry (IS) breaking. I derive the superexchange
interactions for a $t_{2g}$-orbital Hubbard model on a square lattice with the
strong spin-orbit coupling and the IS-breaking-induced hopping integrals, and
explain the microscopic origins of the Dzyaloshinsky-Moriya (DM) -type and the
Kitaev-type interactions. Then, by adopting the mean-field approximation to a
minimal model including only the Heisenberg-type and the DM-type
nearest-neighbor interactions, I show that the IS breaking causes the
spin-orbital-coupled chirality as a result of stabilizing the screw state. I
also highlight the limit of the hard-pseudospin approximation in discussing the
stability of the screw states in the presence of both the DM-type and the
Kitaev-type interactions, and discuss its meaning. I finally discuss the
effects of tetragonal crystal field and $J_{\textrm{eff}}=\frac{3}{2}$ states,
and the application to the iridates near the $[001]$ surface of
Sr$_{2}$IrO$_{4}$ and the interface between Sr$_{2}$IrO$_{4}$ and
Sr$_{3}$Ir$_{2}$O$_{7}$.
