  When the inversion symmetry is broken at a surface, spin-orbit interaction
gives rise to spin-dependent energy shifts - a phenomenon which is known as the
spin Rashba effect. Recently, it has been recognized that an orbital
counterpart of the spin Rashba effect - the orbital Rashba effect - can be
realized at surfaces even without spin- orbit coupling. Here, we propose a
mechanism for the orbital Rashba effect based on sp orbital hybridization,
which ultimately leads to the electric polarization of surface states. As a
proof of principle, we show from first principles that this effect leads to
chiral orbital textures in $\mathbf{k}$-space of the BiAg$_2$ monolayer. In
predicting the magnitude of the orbital moment arising from the orbital Rashba
effect, we demonstrate the crucial role that the Berry phase theory plays for
the magnitude and variation of the orbital textures. As a result, we predict a
pronounced manifestation of various orbital effects at surfaces, and proclaim
the orbital Rashba effect to be a key platform for surface orbitronics.
