  We consider quantum graphs with spin-orbit couplings at the vertices.
Time-reversal invariance implies that the bond S-matrix is in the orthogonal or
symplectic symmetry class, depending on spin quantum number s being integer or
half-integer, respectively. The periodic-orbit expansion of the spectral form
factor is shown to acquire additional weights from spin rotations along orbits.
We determine the spin contribution to the coefficients in an expansion of the
form factor from properties of the representation of the group of spin
transformations on the graph. Consistency with the Circular Orthogonal and
Circular Symplectic Ensemble, respectively, of random matrices is obtained.
