Arrays of nanoscopic magnetic asymmetric rings, 150 nm in outer diameter, are fabricated using the techniques of electron-beam lithography, angular deposition and ion-beam etching. Magnetic measurements for cobalt asymmetric rings at room temperature verifies previous reports of vortex magnetic state formation of a desired circulation direction for the application of external magnetic field along the asymmetry axis of the rings. However, the main theme of this article is the observation of exchange bias phenomena when the ring samples are cooled down to low temperature in the presence of a positive magnetic field. Very interestingly, the observed exchange bias effect is negative for along and perpendicular orientations of ring's asymmetry axis with respect to the in-plane external magnetic field. This is in good quantitative agreement with the random interface model proposed by Malozemoff et al. For the application of in-plane external magnetic field at 451 with respect to the asymmetry axis, the exchange bias effect is positive. Unlike the exchange bias effects in thin films, this is a very unusual observation indicating that exchange bias phenomena of opposite natures can be manipulated by appropriate combinations of geometrical constraint and external magnetic field direction, in addition to the interfacial interactions between ferromagnetic (FM) and antiferromagnetic (AFM) layer.