  We investigate Z2 x Z2 orientifolds with group actions involving shifts. A
complete classification of possible geometries is presented where also previous
work by other authors is included in a unified framework from an intersecting
D-brane perspective. In particular, we show that the additional shifts not only
determine the topology of the orbifold but also independently the presence of
orientifold planes. In the second part, we work out in detail a basis of
homological three cycles on shift Z2 x Z2 orientifolds and construct all
possible fractional D-branes including rigid ones. A Pati-Salam type model with
no open-string moduli in the visible sector is presented.
