In general relativity, freely-falling test objects follow geodesics of the background spacetime in which they live. In a sense, this feature is a mere rephrasing of Einstein’s equivalence principle. In 1968, Brandon Carter showed that the geodesic motion of objects orbiting a Kerr black hole was integrable, in the sense of Hamiltonian mechanics, by discovering a fourth constant of motion that now bears his name. This “universality” of geodesic free fall is, however, but an approximation: In general, two different bodies will follow two distinct paths, depending on how they spin and deform. I will show how, and to which extent, Carter’s integrability can be extended from geodesics to the motion of extended bodies that can spin and deform, and point out how black hole symmetries have, yet again, a special tendency to simplify the problem.