  We study the spin vortices and skyrmions coherently imprinted into an
exciton-polariton condensate on a planar semiconductor microcavity. We
demonstrate that the presence of a polarization anisotropy can induce a complex
dynamics of these structured topologies, leading to the twist of their
circuitation on the Poincar\'e sphere of polarizations. The theoretical
description of the results carries the concept of generalized quantum vortices
in two-component superfluids, which are conformal with polarization loops
around an arbitrary axis in the pseudospin space.
