  The spin-split states subject to Rashba spin-orbit coupling in
two-dimensional systems have long been accepted as pointing inplane and
perpendicular to the corresponding wave vectors. This is in general true for
free electron model, but exceptions do exist elsewhere. Within the
tight-binding model, we unveil the unusual upstanding behavior of those Rashba
spins around $\bar{K}$ and $\bar{K}^{\prime}$ points in honeycomb lattices. Our
calculation (i) explains the recent experiment of the Tl/Si(111)-$(1\times1)$
surface alloy [Phys. Rev. Lett. \textbf{102}, 096805 (2009)], where abrupt
upstanding spin states near $\bar{K}$ are observed, and (ii) predicts an
electrically reversible out-of-plane surface spin polarization.
