  We compute the quotient of the self-duality equation for conformal metrics by
the action of the diffeomorphism group. We also determine Hilbert polynomial,
counting the number of independent scalar differential invariants depending on
the jet-order, and the corresponding Poincar\'e function. We describe the field
of rational differential invariants separating generic orbits of the
diffeomorphism pseudogroup action, resolving the local recognition problem for
self-dual conformal structures.
