We report on the spin properties of bright polariton solitons supported by an external pump to compensate losses. We observe robust circularly polarized solitons when a circularly polarized pump is applied, a result attributed to phase synchronization between nondegenerate TE and TM polarized polariton modes at high momenta. For the case of a linearly polarized pump, either σ+ or σ− circularly polarized bright solitons can be switched on in a controlled way by a σ+ or σ− writing beam, respectively. This feature arises directly from the widely differing interaction strengths between co- and cross-circularly polarized polaritons. In the case of orthogonally linearly polarized pump and writing beams, the soliton emission on average is found to be unpolarized, suggesting strong spatial evolution of the soliton polarization. The observed results are in agreement with theory, which predicts stable circularly polarized solitons and unstable linearly polarized solitons.