  Within the framework of the flux formulation of Double Field Theory (DFT) we
employ a generalised Scherk-Schwarz ansatz and discuss the classification of
the twists that in the presence of the strong constraint give rise to constant
generalised fluxes interpreted as gaugings. We analyse the various
possibilities of turning on the fluxes $H_{ijk}, F_{ij}{}^k, Q_i{}^{jk}$ and
$R^{ijk}$, and the solutions for the twists allowed in each case. While we do
not impose the DFT (or equivalently supergravity) equations of motion, our
results provide solution-generating techniques in supergravity when applied to
a background that does solve the DFT equations. At the same time, our results
give rise also to canonical transformations of 2-dimensional $\sigma$-models, a
fact which is interesting especially because these are integrability-preserving
transformations on the worldsheet. Both the solution-generating techniques of
supergravity and the canonical transformations of 2-dimensional $\sigma$-models
arise as maps that leave the generalised fluxes of DFT and their flat
derivatives invariant. These maps include the known
abelian/non-abelian/Poisson-Lie T-duality transformations, Yang-Baxter
deformations, as well as novel generalisations of them.
