The influence of the optical eigenwaves' polarization in the acousto-optic (AO) interaction in the optically active cubic crystals is analyzed in the present work on the example of cubic, optically active NaClO3 crystals. It has been shown that when the incident optical wave is circularly polarized, the effective elasto-optic (EO) coefficient and AO figure of merit increase almost two times while the surfaces of these parameters undergo inflation compared with those manifested in the interaction of the linearly polarized optical waves. It has been found that six isotropic types of AO interaction are reduced to three types, i.e., the sign of the circular polarization does not affect the relation for the effective EO coefficient. Moreover, in this case, the relations for effective EO coefficients for the anisotropic AO interactions are the same as for isotropic types with respective AWs. We have shown that two approaches we recently developed, based on considering the ellipticity of optical eigenwaves and extraction of the circular polarization from the elliptical polarization state, agree when the diffracted optical wave's polarization is not considered. It has been found that in the presence of circular birefringence in the case when the polarization of the incident optical wave corresponds to the polarization of the eigenwave, the polarization of the diffracted optical wave is not necessarily the same as the polarization of the eigenwave, which corresponds to the phase-matching conditions. In the last case, the polarization of the diffracted wave is determined by the ratio between EO tensor components, which are included in the effective EO coefficient. In general, this polarization is elliptical and not the same as the polarization of the eigenwave.