  We explain how twistor theory represents the self-dual sector of four
dimensional gravity in terms of the loop group of Poisson diffeomorphisms of
the plane via Penrose's non-linear graviton construction. The symmetries of the
self-dual sector are generated by the corresponding loop algebra
$Lw_{1+\infty}$ of the algebra $w_{1+\infty}$ of these Poisson diffeomorphisms.
We show that these coincide with the infinite tower of soft graviton symmetries
in tree-level perturbative gravity recently discovered in the context of
celestial amplitudes. We use a twistor sigma model for the self-dual sector
which describes maps from the Riemann sphere to the asymptotic twistor space
defined from characteristic data at null infinity ${\mathcal I}$. We show that
the OPE of the sigma model naturally encodes the Poisson structure on twistor
space and gives rise to the celestial realization of $Lw_{1+\infty}$. The
vertex operators representing soft gravitons in our model act as currents
generating the wedge algebra of $w_{1+\infty}$ and produce the expected
celestial OPE with hard gravitons of both helicities. We also discuss how the
two copies of $Lw_{1+\infty}$, one for each of the self-dual and anti-self-dual
sectors, are represented in the OPEs of vertex operators of the 4d ambitwistor
string.
