  We give a reformulation of non-linear Einstein gravity, which contains the
dual graviton together with the ordinary metric and a shift gauge field. The
metric does not enter through a `kinetic' Einstein-Hilbert term, but via
topological couplings, and so the theory does not lead to a doubling of degrees
of freedom. The field equations take the form of first-order duality relations.
We analyze the gauge symmetries and comment on their meaning with regard to the
E11 proposal.
