We argue that gravity theories in AdS4 are holographically dual to either of
two three-dimensional CFT's: the usual Dirichlet CFT1 where the fixed graviton
acts as a source for the stress-energy tensor, and a dual CFT2 with a fixed
dual graviton which acts as a source for a dual stress-energy tensor. The dual
stress-energy tensor is shown to be the Cotton tensor of the Dirichlet CFT. The
two CFT's are related by a Legendre transformation generated by a gravitational
Chern-Simons coupling. This duality is a gravitational version of
electric-magnetic duality valid at any radius r, where the renormalized
stress-energy tensor is the electric field and the Cotton tensor is the
magnetic field. Generic Robin boundary conditions lead to CFT's coupled to
Cotton gravity or topologically massive gravity. Interaction terms with CFT1
lead to a non-zero vev of the stress-energy tensor in CFT2 coupled to gravity
even after the source is removed. We point out that the dual graviton also
exists beyond the linearized approximation, and spell out some of the details
of the non-linear construction
