  We study integrable realizations of conformal twisted boundary conditions for
sl(2) unitary minimal models on a torus. These conformal field theories are
realized as the continuum scaling limit of critical G = A,D,E lattice models
with positive spectral parameter u > 0 and Coxeter number g. Integrable seams
are constructed by fusing blocks of elementary local face weights. The usual
A-type fusions are labelled by the Kac labels (r, s) and are associated with
the Verlinde fusion algebra. We introduce a new type of fusion in the two braid
limits u->\pm i\infty associated with the graph fusion algebra, and labelled by
nodes $a,b\in G$ respectively. When combined with automorphisms, they lead to
general integrable seams labelled by $x=(r,a,b,\kappa) \in (A_{g-2}, H, H, Z_2
)$ where H is the graph G itself for Type I theories and its parent for Type II
theories. Identifying our construction labels with the conformal labels of
Petkova and Zuber, we find that the integrable seams are in one-to-one
correspondence with the conformal seams. The distinct seams are thus associated
with the nodes of the Ocneanu quantum graph. The quantum symmetries and twisted
partition functions are checked numerically for $|G| \leq 6$. We also show, in
the case of $D_{2l}$, that the non-commutativity of the Ocneanu algebra of
seams arises because the automorphisms do not commute with the fusions.
