  The problem of finding boundary states in CFT, often rephrased in terms of
"NIMreps" of the fusion algebra, has a natural extension to CFT on
non-orientable surfaces. This provides extra information that turns out to be
quite useful to give the proper interpretation to a NIMrep. We illustrate this
with several examples. This includes a rather detailed discussion of the
interesting case of the simple current extension of A_2 level 9, which is
already known to have a rich structure. This structure can be disentangled
completely using orientation information. In particular we find here and in
other cases examples of diagonal modular invariants that do not admit a NIMrep,
suggesting that there does not exist a corresponding CFT. We obtain the
complete set of NIMreps (plus Moebius and Klein bottle coefficients) for many
exceptional modular invariants of WZW models, and find an explanation for the
occurrence of more than one NIMrep in certain cases. We also (re)consider the
underlying formalism, emphasizing the distinction between oriented and
unoriented string annulus amplitudes, and the origin of orientation-dependent
degeneracy matrices in the latter.
