We examine the symmetry properties of spin-coupled (or full generalised valence bond) wavefunctions for C_{2}H_{2} and N_{2}. The symmetry-separated (σ,π) and bent-bond (ω) solutions are totally symmetric only in the *D*_{4h} and *D*_{3h} subgroups of *D*_{∞}* _{h}*, respectively. Two fairly different strategies are explored for imposing full cylindrical symmetry, with one of them (small nonorthogonal configuration interaction calculations involving rotated versions of the wavefunction) turning out to be somewhat preferable on energetic grounds to the other one (application of additional spin constraints to a single spatial configuration). It is also shown that mixing together the cylindrically symmetric symmetry-separated and bent-bond spin-coupled models leads to relatively small energy improvements unless the valence orbitals in each type of configuration are reoptimised.