  We define generalised chiral vertex operators covariant under the Ocneanu
``double triangle algebra'' {\cal A}, a novel quantum symmetry intrinsic to a
given rational 2-d conformal field theory. This provides a chiral approach,
which, unlike the conventional one, makes explicit various algebraic structures
encountered previously in the study of these theories and of the associated
critical lattice models, and thus allows their unified treatment. The
triangular Ocneanu cells, the 3j-symbols of the weak Hopf algebra {\cal A},
reappear in several guises. With {\cal A} and its dual algebra {hat A} one
associates a pair of graphs, G and {\tilde G}. While G are known to encode
complete sets of conformal boundary states, the Ocneanu graphs {\tilde G}
classify twisted torus partition functions. The fusion algebra of the twist
operators provides the data determining {\hat A}. The study of bulk field
correlators in the presence of twists reveals that the Ocneanu graph quantum
symmetry gives also an information on the field operator algebra.
