I will first review the use of G-structures in the context of supergravity, showing how fluxes can be viewed as intrinsic torsions. This can be used to characterize back-reaction of metric of restricted holonomy once fluxes are introduced. Considering more generally supersymmetry of the effective action, I will argue for a mirror symmetry between manifolds of SU(3) structure that generalize Calabi-Yau manifolds and in the more general framework of the Clifford(d,d) spinors considered recently by Hitchin.