  We consider the conformal group of a space of dim n=p+q, with SO(p,q) metric.
The quotient of this group by its homogeneous Weyl subgroup gives a principal
fiber bundle with 2n-dim base manifold and Weyl fibers. The Cartan
generalization to a curved 2n-dim geometry admits an action functional linear
in the curvatures. Because symmetry is maintained between the translations and
the special conformal transformations in the construction, these spaces are
called biconformal; this same symmetry gives biconformal spaces overlapping
structures with double field theories, including manifest T duality. We
establish that biconformal geometry is a form of double field theory, showing
how general relativity with integrable local scale invariance arises from its
field equations. While we discuss the relationship between biconformal
geometries and the double field theories of T-dual string theories, our
principal interest is the study of the gravity theory. We show that vanishing
torsion and vanishing co-torsion solutions to the field equations overconstrain
the system, implying a trivial biconformal space. Wih co-torsion unconstrained,
we show that (1) the torsion-free solutions are foliated by copies of an n-dim
Lie group, (2) torsion-free solutions generically describe locally
scale-invariant general relativity with symmetric, divergence-free sources on
either the co-tangent bundle of n-dim (p,q)-spacetime or the torus of double
field theory, and (3) torsion-free solutions admit a subclass of spacetimes
with n-dim non-abelian Lie symmetry. These latter cases include the possibility
of a gravity-electroweak unification. It is notable that the field equations
reduce all curvature components to dependence only on the solder form of an
n-dim Lagrangian submanifold, despite the increased number of curvature
components and doubled number of initial independent variables.
