  We analyze unoriented Wess-Zumino-Witten models from a geometrical point of
view. We show that the geometric interpretation of simple current crosscap
states is as centre orientifold planes localized on conjugacy classes of the
group manifold. We determine the locations and dimensions of these planes for
arbitrary simply-connected groups and orbifolds thereof. The dimensions of the
O-planes turn out to be given by the dimensions of symmetric coset manifolds
based on regular embeddings. Furthermore, we give a geometrical interpretation
of boundary conjugation in open unoriented WZW models; it yields D-branes
together with their images under the orientifold projection. To find the
agreement between O-planes and crosscap states, we find explicit answers for
lattice extensions of Gaussian sums. These results allow us to express the
modular P-matrix, which is directly related to the crosscap coefficient, in
terms of characters of the horizontal subgroup of the affine Lie algebra. A
corollary of this relation is that there exists a formal linear relation
between the modular P- and the modular S-matrix.
