  Working in the self-dual sector for Yang-Mills and gravity, we show how to
construct an extended phase space at null infinity, to all orders in the radial
expansion. This formalises the symmetry origin of the infrared behaviour of
these theories to all sub-leading orders. As a corollary, we also derive a
double copy mapping from a subset of YM gauge transformations to a subset of
diffeomorphisms to all orders in the transformation parameters, which to our
knowledge has not been presented before in the literature.
