ATP synthase mediates proton flow through its membrane portion, F0, which drives the synthesis of ATP in its headpiece, F1. The F1-portion contains a hexagonal array of three subunits alpha and three beta encircling a central subunit gamma, that in turn interacts with a smaller epsilon and with F0. Recently we reported that the application of polarized absorption recovery after photobleaching showed the ATP-driven rotation of gamma over at least two, if not three, beta. Here we extend probes of such rotation aided by a new theory for assessing continuous versus stepped, Brownian versus unidirectional molecular motion. The observed relaxation of the absorption anisotropy is fully compatible with a unidirectional and stepping rotation of gamma over three equidistantly spaced angular positions in the hexagon formed by the alternating subunits alpha and beta. The results strongly support a rotational catalysis with equal participation of all three catalytic sites. In addition we report a limited rotation of gamma without added nucleotides, perhaps idling and of Brownian nature, that covers only a narrow angular domain.