  We construct a new covariant action for "flat" self-dual gravity in four
spacetime dimensions. The action has just one term, but when expanded around an
appropriate background gives rise to a kinetic term and a cubic interaction.
Upon imposing the light-cone gauge, the action reproduces the expected chiral
interaction of Siegel. The new action is in many ways analogous to the known
covariant action for self-dual Yang-Mills theory. There is also a sense in
which the new self-dual gravity action exhibits the double copy of self-dual
Yang-Mills structure.
