  We consider orientifold actions involving the permutation of two identical
factor theories. The corresponding crosscap states are constructed in rational
conformal field theory. We study group manifolds, in particular the examples
$SU(2) \times SU(2)$ and $U(1)\times U(1)$ in detail, comparing conformal field
theory results with geometry. We then consider orientifolds of tensor products
of N=2 minimal models, which have a description as coset theories in rational
conformal field theory and as Landau Ginzburg models. In the Landau Ginzburg
language, B-orientifolds and D-branes are described in terms of matrix
factorizations of the superpotential. We match the factorizations with the
corresponding crosscap states.
