  Waveguiding structures made of anisotropic media support bound states in the
continuum (BICs) that arise when the radiation channel of otherwise semi-leaky
modes is suppressed. Hitherto, only structures with optical axes aligned in
symmetric orientations inside the waveguide plane, where BICs appear as lines
in the momentum-frequency dispersion diagram, have been considered. Here we
address settings where such symmetry is broken and unveil a number of
fundamental new features. Weak and strong symmetry-breaking regimes are
identified, corresponding to azimuthal and polar optical axes orientation
asymmetries, respectively. The azimuthal symmetry-breaking is found to still
preserve the existence loci of BICs in the momentum-frequency dispersion
diagram as lines. However, all possible BICs become interferometric, while the
polarization separable states that occur in symmetric settings cease to exist.
The polar symmetry-breaking has stronger effects and transforms the BICs'
existence loci from lines to points, which correspond to full-vector states
that exist at discrete values of the optical axis orientation for a given
wavelength. Such transformation results in fundamental changes in the
topological properties of the radiated field around the BICs.
