Collective (elementary) excitations of quantum bosonic condensates, including condensates of exciton polaritons in semiconductor microcavities, are a sensitive probe of interparticle interactions. In anisotropic microcavities with momentum-dependent transverse-electric-transverse-magnetic splitting of the optical modes, the excitations' dispersions are predicted to be strongly anisotropic, which is a consequence of the synthetic magnetic gauge field of the cavity, as well as the interplay between different interaction strengths for polaritons in the singlet and triplet spin configurations. Here, by directly measuring the dispersion of the collective excitations in a high-density optically trapped exciton-polariton condensate, we observe excellent agreement with the theoretical predictions for spinor polariton excitations. We extract the interaction constants for polaritons of the same and opposite spin and map out the characteristic spin textures in an interacting spinor condensate of exciton polaritons.