A comprehensive understanding of spin-polarized photoemission is crucial for accessing the electronic structure of spin-orbit coupled materials. Yet, the impact of the final state in the photoemission process on the photoelectron spin has been difficult to assess in these systems. We present experiments for the spin-orbit split states in a Bi-Ag surface alloy showing that the alteration of the final state with energy may cause a complete reversal of the photoelectron spin polarization. We explain the effect on the basis of ab initio one-step photoemission theory and describe how it originates from linear dichroism in the angular distribution of photoelectrons. Our analysis shows that the modulated photoelectron spin polarization reflects the intrinsic spin density of the surface state being sampled differently depending on the final state, and it indicates linear dichroism as a natural probe of spin-orbit coupling at surfaces.