The fundamental polarization singularities of monochromatic light are normally associated with invariance under coordinated rotations: symmetry operations that rotate the spatial dependence of an electromagnetic field by an angle *θ* and its polarization by a multiple *γθ* of that angle. These symmetries are generated by mixed angular momenta of the form *J*_{γ} = *L* + *γS*, and they generally induce Möbius-strip topologies, with the coordination parameter *γ* restricted to integer and half-integer values. In this work we construct beams of light that are invariant under coordinated rotations for arbitrary rational *γ*, by exploiting the higher internal symmetry of ‘bicircular’ superpositions of counter-rotating circularly polarized beams at different frequencies. We show that these beams have the topology of a torus knot, which reflects the subgroup generated by the torus-knot angular momentum *J*_{γ}, and we characterize the resulting optical polarization singularity using third- and higher-order field moment tensors, which we experimentally observe using nonlinear polarization tomography.