  We relate the unconstrained `double metric' of the `$\alpha'$-geometry'
formulation of double field theory to the constrained generalized metric
encoding the spacetime metric and b-field. This is achieved by integrating out
auxiliary field components of the double metric in an iterative procedure that
induces an infinite number of higher-derivative corrections. As an application
we prove that, to first order in $\alpha'$ and to all orders in fields, the
deformed gauge transformations are Green-Schwarz-deformed diffeomorphisms. We
also prove that to first order in $\alpha'$ the spacetime action encodes
precisely the Green-Schwarz deformation with Chern-Simons forms based on the
torsionless gravitational connection. This seems to be in tension with
suggestions in the literature that T-duality requires a torsionful connection,
but we explain that these assertions are ambiguous since actions that use
different connections are related by field redefinitions.
