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
