  We recast the action of pure gravity into a form that is invariant under a
twofold Lorentz symmetry. To derive this representation, we construct a general
parameterization of all theories equivalent to the Einstein-Hilbert action up
to a local field redefinition and gauge fixing. We then exploit this freedom to
eliminate all interactions except those exhibiting two sets of independently
contracted Lorentz indices. The resulting action is local, remarkably simple,
and naturally expressed in a field basis analogous to the exponential
parameterization of the nonlinear sigma model. The space of twofold Lorentz
invariant field redefinitions then generates an infinite class of equivalent
representations. By construction, all off-shell Feynman diagrams are twofold
Lorentz invariant while all on-shell tree amplitudes are automatically twofold
gauge invariant. We extend our results to curved spacetime and calculate the
analogue of the Einstein equations. While these twofold invariances are hidden
in the canonical approach of graviton perturbation theory, they are naturally
expected given the double copy relations for scattering amplitudes in gauge
theory and gravity.
