The exchange bias properties of Co/CoO coaxial core/shell nanowires have been investigated with cooling and applied fields perpendicular to the wire axis. This configuration leads to unexpected exchange-bias effects. Firstly, the magnetization value at high fields is found to depend on the field-cooling conditions. This effect arises from the competition between the magnetic anisotropy and the Zeeman energies for cooling fields perpendicular to the wire axis. This allows imprinting pre-defined magnetization states to the AFM, as corroborated by micromagnetic simulations. Secondly, the system exhibits a high-field magnetic irreversibility, leading to open hysteresis loops, attributed to the AFM easy-axis reorientation during the reversal (effect similar to athermal training). A distinct way to manipulate the high-field magnetization in exchange-biased systems, beyond the archetypical effects, is thus experimentally and theoretically demonstrated.