We show the possible existence of a significant axial asymmetry in the reconnection separator at the Jovian magnetopause using first-principle, physics-based global simulations. Under eastward interplanetary magnetic field (IMF) conditions near Jupiter's orbit, reconnection occurs at the southern-dusk and northern magnetopause with large shear angles between the magnetospheric and magnetosheath magnetic fields. When driven by the westward IMF, the reconnection position switches to the northern-dusk and southern magnetopause. Component reconnection at the southern-dusk/northern-dusk magnetopause is associated with the interaction of the IMF with the nearly-dipolar background fields. Nearly-antiparallel reconnection near the noon-midnight plane at the northern/southern magnetopause is related to the dawn-dusk asymmetric, helical, closed lobe magnetic fields, which is a consequence of significant planetary corotation effects and not expected at Earth. Such configuration is testable as Juno has proceeded its orbit to the high-altitude cusps and provides new insight into the interpretation of measurements from other rotationally-driven systems.