  We consider the D1-D5 CFT near the orbifold point, specifically the
computation of correlators involving twist sector fields using covering surface
techniques. As is well known, certain twists introduce spin fields on the
cover. Here we consider the bosonization of fermions to facilitate computations
involving the spin fields. We find a set of cocycle operators that satisfy
constraints coming from various $SU(2)$ symmetries, including the
$SU(2)_L\times SU(2)_R$ R-symmetry. Using these cocycles, we consider the
correlator of four spin fields on the cover, and show that it is invariant
under all of the $SU(2)$ symmetries of the theory. We consider the mutual
locality of operators, and compute several three-point functions. These
computations lead us to a notion of radial ordering on the cover that is
inherited from the original computation before lifting. Further, we note that
summing over orbifold images sets certain branch-cut ambiguous correlators to
zero.
