  We introduce a finite-dimensional algebra that controls the possible boundary
conditions of a conformal field theory. For theories that are obtained by
modding out a Z_2 symmetry (corresponding to a so-called D_odd-type, or
half-integer spin simple current, modular invariant), this classifying algebra
contains the fusion algebra of the untwisted sector as a subalgebra. Proper
treatment of fields in the twisted sector, so-called fixed points, leads to
structures that are intriguingly close to the ones implied by modular
invariance for conformal field theories on closed orientable surfaces.
