Abstract
The twisted boundary conditions and associated partition functions of the conformal
sl(2) A–D–E
models are studied on the Klein bottle and the Möbius strip. The
A–D–E
minimal lattice models give realization to the complete classification of the open descendants of the
sl(2)
minimal theories. We construct the transfer matrices of these lattice models that are
consistent with non-orientable geometries. In particular, we show that in order to realize all
the Klein bottle amplitudes of different crosscap states, not only the topological flip on the
lattice but also the involution in the spin configuration space must be taken into account.
This involution is the 
symmetry of the Dynkin diagrams which corresponds to the simple current of the
Ocneanu algebra.