  A geometry of superspace corresponding to double field theory is developed,
with type II supergravity in D=10 as the main example. The formalism is based
on an orthosymplectic extension OSp(d,d|2s) of the continuous T-duality group.
Covariance under generalised super-diffeomorphisms is manifest. Ordinary
superspace is obtained as a solution of the orthosymplectic section condition.
A systematic study of curved superspace Bianchi identities is performed, and a
relation to a double pure spinor superfield cohomology is established. A
Ramond-Ramond superfield is constructed as an infinite-dimensional
orthosymplectic spinor. Such objects in minimal orbits under the OSp supergroup
("pure spinors") define super-sections.
