We demonstrate magnetic switching between two 360° domain wall vortex states in cobalt nanorings, which are candidate magnetic states for robust and low power magnetoresistive random access memory (MRAM) devices. These 360° domain wall (DW) or 'twisted onion' states can have clockwise or counterclockwise circulation, the two states for data storage. Reliable switching between the states is necessary for any realistic device. We accomplish this switching by applying a circular Oersted field created by passing current through a metal atomic force microscope tip placed at the center of the ring. After initializing in an onion state, we rotate the DWs to one side of the ring by passing a current through the center, and can switch between the two twisted states by reversing the current, causing the DWs to split and meet again on the opposite side of the ring. A larger current will annihilate the DWs and create a perfect vortex state in the rings.