  Based on the structure of a Lie algebroid for non-geometric fluxes in string
theory, a differential-geometry calculus is developed which combines usual
diffeomorphisms with so-called \beta-diffeomorphisms emanating from gauge
symmetries of the Kalb-Ramond field. This allows to construct a bi-invariant
action of Einstein-Hilbert type comprising a metric, a (quasi-)symplectic
structure \beta and a dilaton. As a salient feature, this symplectic gravity
action and the resulting equations of motion take a form which is similar to
the standard action and field equations. Furthermore, the two actions turn out
to be related via a field redefinition reminiscent of the Seiberg-Witten limit.
Remarkably, this redefinition admits a direct generalization to higher-order
\alpha'-corrections and to the additional fields and couplings appearing in the
effective action of the superstring. Simple solutions to the equations of
motion of the symplectic gravity action, including Calabi-Yau geometries, are
discussed.
