Manifestly T-duality covariant worldsheet string models can be constructed by doubling the coordinate fields. We describe the underlying gauge symmetry of a recently proposed Lorentz invariant doubled worldsheet theory that makes half of the worldsheet degrees of freedom redundant. By shifting the Lagrange multiplier, that enforces the gauge fixing condition, the worldsheet action can be cast into various guises. We investigate the renormalization of this theory using a non-linear background/quantum split by employing a normal coordinate expansion adapted to the gauge-fixed theory. The propagator of the doubled coordinates contains a projection operator encoding that half of them do not propagate. We determine the doubled target space equations of motion by requiring one-loop Weyl invariance. Some of them are generalizations of the conventional sigma model beta-functions, while others seem to be novel to the doubled theory: in particular, a dilaton equation seems related to the strong constraint of double field theory. However, the other target space field equations are not identical to those of double field theory.