Arrays of asymmetric Permalloy nanorings with a decentered inner hole are studied by broadband inductive spectroscopy in the gigahertz range. The spectra display a set of characteristic spin-wave eigenmodes. We find that these eigenmodes probe the absolute vortex circulation direction of a ring. The eigenmode spectra and the signal strengths provide evidence that the asymmetric rings in the large array exhibit an ordered vortex circulation state. The degree of ordering is better than 97%. Micromagnetic simulations reveal that the circulation control is achieved by suppression of domain-wall propagation as the switching mechanism. The deterministic vortex circulation is an important prerequisite for the application of rings in magnetoelectronics.