Above a few tens of MeV per nucleon, large, gradual solar energetic particle (SEP) events are highly variable in their spectral characteristics and elemental composition. The origin of this variability has been a matter of intense and ongoing debate. In this paper, we propose that this variability arises from the interplay of two factors—shock geometry and a compound seed population, typically comprising both solar-wind and flare suprathermals. Whereas quasi-parallel shocks generally draw their seeds from solar-wind suprathermals, quasi-perpendicular shocks—by requiring a higher initial speed for effective injection—preferentially accelerate seed particles from flares. Solar-wind and flare seed particles have distinctive compositional characteristics, which are then reflected in the accelerated particles. We first examine our hypothesis in the context of particles locally accelerated near 1 AU by traveling interplanetary shocks. We illustrate the implications of our hypothesis for SEPs with two very large events, 2002 April 21 and 2002 August 24. These two events arise from very similar solar progenitors but nevertheless epitomize extremes in high-energy SEP variability. We then test our hypothesis with correlation studies based on observations of 43 large SEP events in 1997-2003 by the Advanced Composition Explorer, Wind, the Interplanetary Monitoring Platform 8, and GOES. We consider correlations among high-energy Fe/O, event size, spectral characteristics, the presence of GeV protons, and event duration at high energies. The observed correlations are all qualitatively consistent with our hypothesis. Although these correlation studies cannot be construed as proof of our hypothesis, they certainly confirm its viability. We also examine the alternative hypothesis in which a direct flare component—rather than flare particles subsequently processed through a shock—dominates at high energies. This alternative would produce compositional characteristics similar to those of our hypothesis. However, the observed longitude distribution of the enhanced Fe/O events, their spectral characteristics, and recent timing studies all pose serious challenges for a direct flare component. We also comment on measurements of the mean ionic charge state of Fe at high energies. We conclude that shock geometry and seed population potentially provide a framework for understanding the overall high-energy variability in large SEP events. We suggest additional studies for testing this hypothesis.