  We study the band structure of the $\text{Bi}_2\text{Se}_3$ topological
insulator (111) surface using angle-resolved photoemission spectroscopy. We
examine the situation where two sets of quantized subbands exhibiting different
Rashba spin-splitting are created via bending of the conduction (CB) and the
valence (VB) bands at the surface. While the CB subbands are strongly Rashba
spin-split, the VB subbands do not exhibit clear spin-splitting. We find that
CB and VB experience similar band bending magnitudes, which means, a
spin-splitting discrepancy due to different surface potential gradients can be
excluded. On the other hand, by comparing the experimental band structure to
first principles LMTO band structure calculations, we find that the strongly
spin-orbit coupled Bi 6$p$ orbitals dominate the orbital character of CB,
whereas their admixture to VB is rather small. The spin-splitting discrepancy
is, therefore, traced back to the difference in spin-orbit coupling between CB
and VB in the respective subbands' regions.
