  Based on ab initio relativistic ${\mathbf k}\cdot{\mathbf p}$ theory, we
derive an effective two-band model for surface states of three-dimensional
topological insulators up to seventh order in $\mathbf{k}$. It provides a
comprehensive description of the surface spin structure characterized by a
non-orthogonality between momentum and spin. We show that the oscillation of
the non-orthogonality with the polar angle of $\mathbf{k}$ with a $\pi/3$
periodicity can be seen as due to effective six-fold symmetric spin-orbit
magnetic fields with a quintuple and septuple winding of the field vectors per
single rotation of $\mathbf{k}$. Owing to the dominant effect of the classical
Rashba field, there remains a single-winding helical spin structure but with a
periodic few-degree deviation from the orthogonal locking between momentum and
spin.
