  We investigate the lifting of half-maximal four-dimensional gauged
supergravities to compactifications of string theory. It is shown that a class
of such supergravities can arise from compactifications of IIA string theory on
manifolds of SU(2)-structure which may be thought of as K3 fibrations over T^2.
Examples of these SU(2)-structure backgrounds, as smooth K3 bundles and as
compactifications with H-flux, are given and we also find evidence for a class
of non-geometric, Mirror-fold backgrounds. By applying the duality between IIA
string theory on K3 and Heterotic string theory on T^4 fibrewise, we argue that
these SU(2)-structure backgrounds are dual to Heterotic compactifications on a
class T^4 fibrations over T^2. Examples of these fibrations as twisted tori,
H-flux and T-fold compactifications are given. We also construct a new set of
backgrounds, particular to Heterotic string theory, which includes a previously
unknown class of Heterotic T-folds. A sigma model description of these
backgrounds, from the Heterotic perspective, is presented in which we
generalize the Bosonic doubled formalism to Heterotic string theory.
