  The section condition in double field theory has been shown to imply that a
physical point should be one-to-one identified with a gauge orbit in the
doubled coordinate space. Here we show the converse is also true, and continue
to explore the idea of `spacetime being doubled yet gauged'. Introducing an
appropriate gauge connection, we construct a string action, with an arbitrary
generalized metric, which is completely covariant with respect to the
coordinate gauge symmetry, generalized diffeomorphisms, world-sheet
diffeomorphisms, world-sheet Weyl symmetry and O(D,D) T-duality. A topological
term previously proposed in the literature naturally arises and a self-duality
condition follows from the equations of motion. Further, the action may couple
to a T-dual background where the Riemannian metric becomes everywhere singular.
