Density functional theory (DFT) has been applied to three open-shell molecules (NO₃, O₄⁺, and O₂⁺) for which the unrestricted Hartree-Fock (UHF) wavefunction breaks spatial symmetry. In contrast to Hartree-Fock, all of the standard DFT methods we employed yielded symmetric densities for each of the molecules considered. Symmetry-broken solutions were obtained with DFT only when we used hybrid functionals including unusually large fractions of Hartree-Fock exchange. The exchange functional seems more important than the correlation functional in determining whether symmetry is preserved or broken.