  We demonstrate by breaking the $C_{6}$ symmetry for higher-order at-$\Gamma$
bound states in the continuum (BICs) with topological charge $-2$ in photonic
crystals, (i) deterministic generation of off-$\Gamma$ BICs from the
at-$\Gamma$ BIC, and (ii) a variety of pair-creation and annihilation processes
of circularly-polarized states with opposite topological charge and same
handedness. These processes are well explained by the conservation of
handedness-wise topological charges, showing topologically robustness of these
phenomena. The results indicate a new control of topological aspect of
far-field polarization vectors.
