  We present a global construction of a so-called D-bracket appearing in the
physics literature of Double Field Theory (DFT) and show that if certain
integrability criteria are satisfied, it can be seen as a sum of two Courant
algebroid brackets. In particular, we show that the local picture of the
extended space-time used in DFT fits naturally in the geometrical framework of
para-Hermitian manifolds and that the data of an (almost) para-Hermitian
manifold is sufficient to construct the D-bracket. Moreover, the twists of the
bracket appearing in DFT can be interpreted in this framework geometrically as
a consequence of certain deformations of the underlying para-Hermitian
structure.
