  We argue that M-theory/massive IIA backgrounds including KK-monopoles are
suitably described in the language of G-structures and their intrinsic torsion.
To this end, we study classes of minimal supergravity models that admit an
interpretation as twisted reductions in which the twist parameters are not
restricted to satisfy the Jacobi constraints $\omega\, \omega=0$ required by an
ordinary Scherk-Schwarz reduction. We first derive the correspondence between
four-dimensional data and torsion classes of the internal space and, then,
check the one-to-one correspondence between higher-dimensional and
four-dimensional equations of motion. Remarkably, the whole construction holds
regardless of the Jacobi constraints, thus shedding light upon the
string/M-theory interpretation of (smeared) KK-monopoles.
