  We study the mutual consistency of twisted boundary conditions in the coset
conformal field theory G/H. We calculate the overlap of the twisted boundary
states of G/H with the untwisted ones, and show that the twisted boundary
states are consistently defined in the diagonal modular invariant. The overlap
of the twisted boundary states is expressed by the branching functions of a
twisted affine Lie algebra. As a check of our argument, we study the diagonal
coset theory so(2n)_1 \oplus so(2n)_1/so(2n)_2, which is equivalent with the
orbifold S^1/\Z_2. We construct the boundary states twisted by the
automorphisms of the unextended Dynkin diagram of so(2n), and show their mutual
consistency by identifying their counterpart in the orbifold. For the triality
of so(8), the twisted states of the coset theory correspond to neither the
Neumann nor the Dirichlet boundary states of the orbifold and yield the
conformal boundary states that preserve only the Virasoro algebra.
