  We study the action of picture-changing and spectral flow operators on a
ground ring of ghost number zero operators in the chiral BRST cohomology of the
closed N=2 string and describe an infinite set of symmetry charges acting on
physical states. The transformations of physical string states are compared
with symmetries of self-dual gravity which is the effective field theory of the
closed N=2 string. We derive all infinitesimal symmetries of the self-dual
gravity equations in 2+2 dimensional spacetime and introduce an infinite
hierarchy of commuting flows on the moduli space of self-dual metrics. The
dependence on moduli parameters can be recovered by solving the equations of
the SDG hierarchy associated with an infinite set of abelian symmetries
generated recursively from translations. These non-local abelian symmetries are
shown to coincide with the hidden abelian string symmetries responsible for the
vanishing of most scattering amplitudes. Therefore, N=2 string theory
"predicts" not only self-dual gravity but also the SDG hierarchy.
