The different shapes of spin textures as a journey through Brillouin zone chiral and polar symmetries.

Crystallographic space group symmetry (CPGS) such as polar and nonpolar crystal classes has long been known to classify compounds that have spin-orbit-induced spin splitting such as Rashba and Dresselhaus compounds, respectively. However, the enabling symmetries underlying the spin texture prototypes - the relationship between the expectation value of spin operator $S^{nk}$ in Bloch state $u(n,k)$ and the wavevector $\vec{k}$ - are instead specific rotations and reflection symmetries in the wavevector point group symmetry rather than the presence or absence of polar field in the global crystallographic space group. Relativistic band structures calculations demonstrate how compounds judged by their macroscopic CPGS to be typical Dresselhaus (GaAs) or Rashba (GeTe) or Wyle (Tellurium) compounds, manifest both Rashba and Dresselhaus spin textures, respectively at different wavevector chiralities and polarities.