  Surface electronic structure and its one-dimensionality above and below the
Fermi level ($E_{\rm F}$) were surveyed on the Bi/GaSb(110)-(2$\times$1)
surface hosting quasi-one-dimensional (Q1D) Bi chains, using conventional
(one-photon) and two-photon angle-resolved photoelectron spectroscopy (ARPES)
and theoretical calculations. ARPES results reveal that the Q1D electronic
states are within the projected bulk bandgap. Circular dichroism of two-photon
ARPES and density-functional-theory calculation indicate clear spin and orbital
polarization of the surface states consistent with the giant sizes of
Rashba-type SOI, derived from the strong contribution of heavy Bi atoms. The
surface conduction band above $E_{\rm F}$ forms a nearly straight
constant-energy contour, suggesting its suitability for application in further
studies of one-dimensional electronic systems with strong SOI. A tight-binding
model calculation based on the obtained surface electronic structure
successfully reproduces the surface band dispersions and predicts possible one-
to two-dimensional crossover in the temperature range of 60--100~K.
