  Using a ``Superstrings with Torsion'' type description, we study a class of
IIB orientifolds in which spacefilling O5 planes and D5 branes wrap the T^2
fiber in a warped modification of the product of 4D Minkowski space and a T^2
fibration. For the case that the base is T^4, we provide examples that preserve
4D N = 1, 2, and 3 supersymmetry, both with internal RR flux, and with a
combination of internal RR and NS flux. In these examples, the internal
geometries admit integrable complex structure; however, the almost complex
structure selected by the supersymmetry conditions is nonintegrable in the case
that there is NS flux. We indicate explicitly the massless spectrum of gauge
fields and moduli in each example. In a previous investigation, this class of
orientifolds was studied using T-duality. Here, we extend the previous
analysis, first by providing an intrinsic description that does not rely on
duality, and then by elaborating on details of the T-duality map, which we use
to check our results.
