  Corresponding to the Bondi-Metzner-Sachs (BMS) symmetry algebra of
asymptotically-flat spacetimes are a set of BMS charges. These are formally
constructed via the symplectic formalism of Wald and Zoupas, but the same
charge expression may be arrived at by the simpler twistorial procedure of Dray
and Streubel. Here, we formalize the connection between twistors and asymptotic
symmetries which underlies the Dray-Streubel charge by demonstrating an
isomorphism between twistors in flat spacetime and twistors on radiation-free
sections of $\mathscr{I}^+$. In the corresponding formalism, the Dray-Streubel
charge finds a natural reinterpretation as exactly the part of Penrose's
twistorial charge which is invariant with respect to a certain gauge
transformation. Furthermore, we argue that the twistorial picture of the
radiative phase space, properly formalized, provides a tool alongside the
symplectic formalism or shear structure for analyzing radiative data on
$\mathscr{I}^+$.
