  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.
