We experimentally investigated the switching from onion to vortex states in asymmetric cobalt nanorings by an applied circular field. An in-plane field is applied along the symmetric or asymmetric axis of the ring to establish domain walls (DWs) with symmetric or asymmetric polarization. A circular field is then applied to switch from the onion state to the vortex state, moving the DWs in the process. The asymmetry of the ring leads to different switching fields depending on the location of the DWs and direction of applied field. For polarization along the asymmetric axis, the field required to move the DWs to the narrow side of the ring is smaller than the field required to move the DWs to the larger side of the ring. For polarization along the symmetric axis, establishing one DW in the narrow side and one on the wide side, the field required to switch to the vortex state is an intermediate value.