Recent breakthroughs in numerical relativity have allowed simulations of all three stages (inspiral, plunge, ringdown) of binary black hole (BBH) merger. These simulations have yielded many surprising results, including recoil velocities for the final black hole approaching 4,000 km/s for certain initial spin configurations. We seek to understand the spin dependence of BBH merger by regarding it as a map from a simple initial state (two well separated Kerr black holes) to a simple final state (a single recoiling Kerr black hole). By Taylor expanding this map in the initial BBH spin components and systematically applying symmetry constraints, we obtain a formalism that is simple, yet remarkably successful at explaining existing simulations. Our formalism also makes a host of detailed quantitative predictions for future simulations, and suggests a much more efficient way of mapping out the parameter space of BBH mergers. Since we rely on symmetry rather than dynamics, our expansion remains valid through all stages of the merger (inspiral, plunge, ringdown), and hence complements previous analytical techniques like the post-Newtonian approximation.