Non-linear parent action and dual gravity

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.