  When string theory is compactified on a six-dimensional manifold with a
nontrivial NS flux turned on, mirror symmetry exchanges the flux with a purely
geometrical composite NS form associated with lack of integrability of the
complex structure on the mirror side. Considering a general class of
T^3-fibered geometries admitting SU(3) structure, we find an exchange of pure
spinors (e^{iJ} and \Omega) in dual geometries under fiberwise T-duality, and
study the transformations of the NS flux and the components of intrinsic
torsion. A complementary study of action of twisted covariant derivatives on
invariant spinors allows to extend our results to generic geometries and
formulate a proposal for mirror symmetry in compactifications with NS flux.
