The acousto-optical diffraction of Bessel light beams (BLB) propagating near the optical axis of uniaxial gyrotropic crystals is examined. It is shown that polarization-independent modulation of Bessel beams is possible in paratellurite crystals; that is, the Bragg diffraction efficiency is independent of the polarization state of the incident beam. The physical reason for this kind of modulation is found to be the simultaneous occurrence of two anisotropic diffraction processes, such that the conditions for Bragg synchrony are satisfied for orthogonally polarized elliptical Bessel beams. The coupled wave equations and an overlap integral method are used to study the dependence of the diffraction efficiency on the length of the acousto-optical interaction, the ultrasonic power, and the polarization state of the incident Bessel beam.