  We show that the Teukolsky connection, which defines generalized wave
operators governing the behavior of massless fields on Einstein spacetimes of
Petrov type D, has its origin in a distinguished conformally and GHP covariant
connection on the conformal structure of the spacetime. The conformal class has
a (metric compatible) integrable almost-complex structure under which the
Einstein space becomes a complex (Hermitian) manifold. There is a unique
compatible Weyl connection for the conformal structure, and it leads to the
construction of a conformally covariant GHP formalism and a generalization of
it to weighted spinor/tensor fiber bundles. In particular, `weighted Killing
spinors', previously defined with respect to the Teukolsky connection, are
shown to have their origin in the GHP-Weyl connection, and we show that the
type D principal spinors are actually parallel with respect to it. Furthermore,
we show that the existence of a conformal Killing-Yano tensor can be thought to
be a consequence of the presence of a Kahler metric in the conformal class.
These results provide an interpretation of the persistent hidden symmetries
appearing in black hole perturbations. We also show that the preferred Weyl
connection allows a natural injection of spinor fields into local twistor space
and that this leads to the notion of weighted local twistors. Finally, we find
conformally covariant operator identities for massless fields and the
corresponding wave equations.
