We find that Polyakov lines, computed in abelian-projected SU(2) lattice gauge theory in the confined phase, have finite expectation values for lines corresponding to two units of the abelian electric charge. This means that even multiples of abelian electric charge are unconfined, and that the abelian-projected lattice has at most ₂, rather than U(1), global symmetry. We also find a severe breakdown of the monopole dominance approximation, as well as positivity, in this charge-2 case. Our results on global ₂ symmetry contradict both the dual-superconductor and monopole Coulomb gas pictures of confinement, where all multiples of abelian electric charge (suitably identified in an abelian-projection gauge), should be confined. The breakdown of monopole dominance is also incompatible with a monopole Coulomb gas. Our data is, however, consistent with a center vortex vacuum structure. Further evidence is provided, in lattice Monte Carlo simulations, for collimation of confining color-magnetic flux into vortices.