  We give a double copy construction for the symmetries of the self-dual
sectors of Yang-Mills (YM) and gravity, in the light-cone formulation. We find
an infinite set of double copy constructible symmetries. We focus on two
families which correspond to the residual diffeomorphisms on the gravitational
side. For the first one, we find novel non-perturbative double copy rules in
the bulk. The second family has a more striking structure, as a
non-perturbative gravitational symmetry is obtained from a perturbatively
defined symmetry on the YM side.
  At null infinity, we find the YM origin of the subset of extended
Bondi-Metzner-Sachs (BMS) symmetries that preserve the self-duality condition.
In particular, holomorphic large gauge YM symmetries are double copied to
holomorphic supertranslations. We also identify the single copy of
superrotations with certain non-gauge YM transformations that to our knowledge
have not been previously presented in the literature.
