  We construct a conformally invariant vector bundle connection such that its
equation of parallel transport is a first order system that gives a
prolongation of the conformal Killing equation on differential forms. Parallel
sections of this connection are related bijectively to solutions of the
conformal Killing equation. We construct other conformally invariant
connections, also giving prolongations of the conformal Killing equation, that
bijectively relate solutions of the conformal Killing equation on $k$-forms to
a twisting of the conformal Killing equation on (k - l)-forms for various
integers l. These tools are used to develop a helicity raising and lowering
construction in the general setting and on conformally Einstein manifolds.
