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The ice shell on Enceladus, an icy moon of Saturn, exhibits strong asymmetry between the northern and southern hemispheres, with all known geysers concentrated over the south pole, even though the expected pattern of tidal forced deformation should be symmetric between the north and south poles. Using an idealized ice-evolution model, we demonstrate that this asymmetry may form spontaneously, without any noticeable a priori asymmetry (such as a giant impact or a monopole structure of geological activity), in contrast to previous studies. Infinitesimal asymmetry in the ice shell thickness due to random perturbations are found to be able to grow indefinitely, ending up significantly thinning the ice shell at one of the poles, thereby allowing fracture formation there. Necessary conditions to trigger this hemispheric symmetry-breaking mechanism are found analytically. A rule of thumb we find is that, for Galilean and Saturnian icy moons, the ice shell can undergo hemispheric symmetry breaking only if the mean shell thickness is around 10 to 30 km.