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We simulate the hypothesised collision between the proto-Earth and a Mars-sized impactor that created the Moon. Amongst the resulting debris disk in some impacts, we find a self-gravitating clump of material. It is roughly the mass of the Moon, contains $\sim1\%$ iron like the Moon, and has its internal composition resolved for the first time. The clump contains mainly impactor material near its core but becomes increasingly enriched in proto-Earth material near its surface. A graduated composition has recently been measured in the oxygen isotope ratios of Apollo samples, suggesting incomplete mixing between proto-Earth and impactor material that formed the Moon. However, the formation of the Moon-sized clump depends sensitively on the spin of the impactor. To explore this, we develop a fast method to construct models of multi-layered, rotating bodies and their conversion into initial conditions for smoothed particle hydrodynamical (SPH) simulations. We use our publicly available code to calculate density and pressure profiles in hydrostatic equilibrium, then generate configurations of over a billion particles with SPH densities within $1\%$ of the desired values. This algorithm runs in a few minutes on a desktop computer, for $10^7$ particles, and allows direct control over the properties of the spinning body. In comparison, relaxation or spin-up techniques that take hours on a supercomputer before the structure of the rotating body is even known. Collisions that differ only in the impactor's initial spin reveal a wide variety of outcomes: a merger, a grazing hit-and-run, or the creation of an orbiting proto-Moon.