  Various structural properties of the space of symmetry breaking boundary
conditions that preserve an orbifold subalgebra are established. To each such
boundary condition we associate its automorphism type. It is shown that
correlation functions in the presence of such boundary conditions are
expressible in terms of twisted boundary blocks which obey twisted Ward
identities. The subset of boundary conditions that share the same automorphism
type is controlled by a classifying algebra, whose structure constants are
shown to be traces on spaces of chiral blocks. T-duality on boundary conditions
is not a one-to-one map in general. These structures are illustrated in a
number of examples. Several applications, including the construction of non-BPS
boundary conditions in string theory, are exhibited.
