  We find necessary and sufficient conditions for gauge invariance of the
action of Double Field Theory (DFT) as well as closure of the algebra of gauge
symmetries. The so-called weak and strong constraints are sufficient to satisfy
them, but not necessary. We then analyze compactifications of DFT on twisted
double tori satisfying the consistency conditions. The effective theory is a
Gauged DFT where the gaugings come from the duality twists. The action,
bracket, global symmetries, gauge symmetries and their closure are computed by
twisting their analogs in the higher dimensional DFT. The non-Abelian heterotic
string and lower dimensional gauged supergravities are particular examples of
Gauged DFT.
