Summary: 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.