  Recently it has been shown that D-branes in orientifolds are not always
described by equivariant Real K-theory. In this paper we define a previously
unstudied twisted version of equivariant Real K-theory which gives the D-brane
spectrum for such orientifolds. We find that equivariant Real K-theory can be
twisted by elements of a generalised group cohomology. This cohomology
classifies all orientifolds just as group cohomology classifies all orbifolds.
As an example we consider the $\Omega\times\I_4$ orientifolds. We completely
determine the equivariant orthogonal K-theory $KO_{\Zop_2}(\R^{p,q})$ and
analyse the twisted versions. Agreement is found between K-theory and Boundary
Confromal Field Theory (BCFT) results for both integrally- and torsion-charged
D-branes.
