The Tl/Si(111)-(1 × 1) surface features outstanding properties with a unique spin pattern in momentum space. Along¯ K (¯ K ), surface states with rotating spin had been observed. Here, the focus is put on the unoccupied surface electronic structure along the¯ M high-symmetry direction. Spin-and angle-resolved inversephotoemission experiments with sensitivity to the in-plane and out-of-plane components of the spin-polarization vector are conducted with the help of our recently developed rotatable spin-polarized electron source (ROSE). In agreement with our theoretical findings, we identify a surface resonance with giant spin-orbit-induced spin splitting, which exhibits an exclusive Rashba-type spin texture, i.e., a pure in-plane spin polarization perpendicular to¯ M , in compliance with the mirror-plane symmetry of the¯ M direction. Notably, for slight deviations from the high-symmetry line, this constraint is lifted and additional spin-polarization components emerge. This highlights the importance of a correct alignment of the experiment when investigating samples with spin textures that are more complex than in simple Rashba systems or in ferromagnets.