  We discuss geometry underlying orientifolds with non-trivial NS-NS B-flux. If
D-branes wrap a torus with B-flux the rank of the gauge group is reduced due to
non-commuting Wilson lines whose presence is implied by the B-flux. In the case
of D-branes transverse to a torus with B-flux the rank reduction is due to a
smaller number of D-branes required by tadpole cancellation conditions in the
presence of B-flux as some of the orientifold planes now have the opposite
orientifold projection. We point out that T-duality in the presence of B-flux
is more subtle than in the case with trivial B-flux, and it is precisely
consistent with the qualitative difference between the aforementioned two
setups. In the case where both types of branes are present, the states in the
mixed (e.g., 59) open string sectors come with a non-trivial multiplicity,
which we relate to a discrete gauge symmetry due to non-zero B-flux, and
construct vertex operators for the the mixed sector states. Using these results
we revisit K3 orientifolds with B-flux (where K3 is a T^4/Z_M orbifold) and
point out various subtleties arising in some of these models. For instance, in
the Z_2 case the conformal field theory orbifold does not appear to be the
consistent background for the corresponding orientifolds with B-flux. This is
related to the fact that non-zero B-flux requires the presence of both O5^- as
well as O5^+ planes at various Z_2 orbifold fixed points, which appears to be
inconsistent with the presence of the twisted B-flux in the conformal field
theory orbifold. We also consider four dimensional N=2 and N=1 supersymmetric
orientifolds. We construct consistent four dimensional models with B-flux which
do not suffer from difficulties encountered in the K3 cases.
