We investigate the effect of a deliberately introduced shape asymmetry on magnetization reversal in small, square-shaped, magnetic rings. The magnetization reversal process is investigated using the diffracted magneto-optical Kerr effect combined with micromagnetic simulations. Experimentally we find that the reversal path is sensitive to small (+/-1°) changes in the direction of the applied field. Micromagnetic simulations that reproduce the measured zeroth- and first-order loops allow us to identify the reversal mechanisms as due to different intermediate states, namely, the so-called vortex and horseshoe states. Based on our results we are now able to prescribe a methodology for writing a vortex state with specific chirality in these asymmetric rings. Such control will be necessary if patterned arrays of this kind are to be used as magnetic storage elements.