Eigenmodes characterized by distinct polarization states for the resonator configuration known as the High-Extraction Decentered Annular Ring Resonator (HEXDARR) have been found. The effects of internal annular optical elements (waxicons, reflaxicons, and cones) as well as coatings and mode rotation are included. For a good design, the low-order mode polarization states are nearly circular, with the polarization ellipse orientation and phase angle slightly spatially dependent across the near field of the resonator output. The two lowest-loss eigenmodes are separated by only a small loss which is dependent on the coating properties, one being principally right-hand circularly polarized and the other principally left-hand circularly polarized.