  We consider the su(2) and su(3) affine theories on a cylinder, from the point
of view of their discrete internal symmetries. To this end, we adapt the usual
treatment of boundary conditions leading to the Cardy equation to take the
symmetry group into account. In this context, the role of the Ishibashi states
from all (non periodic) bulk sectors is emphasized. This formalism is then
applied to the su(2) and su(3) models, for which we determine the action of the
symmetry group on the boundary conditions, and we compute the twisted partition
functions. Most if not all data relevant to the symmetry properties of a
specific model are hidden in the graphs associated with its partition function,
and their subgraphs. A synoptic table is provided that summarizes the many
connections between the graphs and the symmetry data that are to be expected in
general.
