  We revisit the problem of constructing type IIA orientifolds on T^6/(Z2 x Z2)
which admit (non)-factorisable lattices. More concretely, we consider a (Z2 x
Z2') orientifold with torsion, where D6-branes wrap rigid 3-cycles. We derive
the model building rules and consistency conditions in the case where the
compactification lattice is non-factorisable. We show that in this class of
configurations, (semi) realistic models with an odd number of families can be
easily constructed, in contrast to compactifications where the D6-branes wrap
non-rigid cycles. We also show that an odd number of families can be obtained
in the factorisable case, without the need of tilted tori. We illustrate the
discussion by presenting three family Pati-Salam models with no chiral exotics
in both factorisable and non-factorisable toroidal compactifications.
