  Recently, we introduced the "Newman-Penrose map," a novel correspondence
between a certain class of solutions of Einstein's equations and self-dual
solutions of the vacuum Maxwell equations, which we showed was closely related
to the classical double copy. Here, we give an alternative definition of this
correspondence in terms of quantities that are naturally defined on both
spacetime and twistor space. The advantage of this reformulation is that it is
purely geometrical in nature, being manifestly invariant under both spacetime
diffeomorphisms and projective transformations on twistor space. While the
original formulation of the map may be more convenient for most explicit
calculations, the twistorial formulation we present here may be of greater
theoretical utility.
