  The geometry of N=1 supersymmetric double field theory is revisited in
superspace. In order to maintain the constraints on the torsion tensor, the
local tangent space group of O(D) x O(D) must be expanded to include a tower of
higher dimension generators. These include a generator in the irreducible hook
representation of the Lorentz group, which gauges the shift symmetry (or
ambiguity) of the spin connection. This gauging is possible even in the purely
bosonic theory, where it leads to a Lorentz curvature whose only non-vanishing
pieces are the physical ones: the generalized Einstein tensor and the
generalized scalar curvature. A relation to the super-Maxwell$_\infty$ algebra
is proposed. The superspace Bianchi identities are solved up through dimension
two, and the component supersymmetry transformations and equations of motion
are explicitly (re)derived.
