Most ab initio calculations on fullerene molecules have been carried out on the basis of the paradigm of the Huckel model. This is consistent with the restricted nature of the independent-particle model underlying such calculations, even in single-reference-based correlated approaches. Notwithstanding, previous works on some of these molecules using model Hamiltonians have clearly indicated the importance of short-range interatomic spin-spin correlations. In this work, we consider ab initio non-collinear Hartree-Fock (HF) solutions for representative fullerene systems: the bowl, cage, ring, and pentagon isomers of C-20, and the larger C-30, C-36, C-60, C-70, and C-84 fullerene cages. In all cases but the ring we find that the HF minimum corresponds to a truly non-collinear solution with a torsional spin density wave. Optimized geometries at the generalized HF (GHF) level lead to fully symmetric structures, even in those cases where Jahn-Teller distortions have been previously considered. The nature of the GHF solutions is consistent with the p-electron space becoming polyradical in nature: each p-orbital remains effectively singly occupied. The spin frustration, induced by the presence of pentagon rings on an otherwise antiferromagnetic background, is minimized at the HF level by aligning the spins with non-collinear arrangements. The long-range magnetic ordering observed is reminiscent of the character of broken symmetry HF solutions in polyacene systems.