  We investigate the symmetry algebra of the recently proposed field theory on
a doubled torus that describes closed string modes on a torus with both
momentum and winding. The gauge parameters are constrained fields on the
doubled space and transform as vectors under T-duality. The gauge algebra
defines a T-duality covariant bracket. For the case in which the parameters and
fields are T-dual to ones that have momentum but no winding, we find the gauge
transformations to all orders and show that the gauge algebra reduces to one
obtained by Siegel. We show that the bracket for such restricted parameters is
the Courant bracket. We explain how these algebras are realised as symmetries
despite the failure of the Jacobi identity.
