  We perform a Kaluza-Klein inspired rewriting of double field theory by
splitting the coordinates into `compact' and `non-compact' directions. There is
no truncation of the compact coordinates or their duals, and so this
formulation is manifestly O(d,d) invariant, with d the number of compact
directions. The action can serve as starting point for arbitrary Kaluza-Klein
ansaetze. For a torus background the theory describes the full tower of
Kaluza-Klein modes or, in the dual frame, of the winding modes. The
Kaluza-Klein vector is a gauge field for the duality-covariantized Courant
bracket algebra rather than a Lie algebra. Gauge covariance requires the
inclusion of the 2-form gauge potential descending from the Kalb-Ramond field,
leading to a structure resembling the tensor hierarchy of gauged supergravity.
