The energy spectra of Fe in the very large solar energetic particle (SEP) event of 2000 July 14 are strikingly different from those of lighter species. We show that this difference can be explained by shock acceleration from a two-component source population, comprising solar wind suprathermals and a small (∼5%) admixture of remnant flare particles, as previously proposed to explain enhanced ^3He/^4He in some gradual SEP events. Flare remnants can also account for several previously unexplained features of high-energy solar heavy ions as well as important aspects of SEP event-to-event variability. These results offer a new perspective on the enduring controversy over the relative roles of flares and coronal mass ejections (CMEs) in producing SEPs. Flare activity clearly makes a unique and critical contribution to the source population. But the predominate accelerator in large gradual SEP events is the CME-driven shock, and many spectral, compositional, and charge state characteristics of highenergy heavy ions can be understood without invoking other acceleration mechanisms.