Ice is a very complex and fundamentally important solid. In the present article, we review a new property of the hydrogen-bonded network in ice structures: an explicit nonequivalence of some antipodal configurations with the opposite direction of all hydrogen bonds (H-bonds). This asymmetry is most pronounced for the structures with considerable deviation of the H-bond network from the tetrahedral coordination. That is why we have investigated in detail four-coordinated ice nanostructures with no outer dangling hydrogen atoms, namely, ice bilayers and ice nanotubes consisting of stacked n-membered rings. The reason for this H-bonding asymmetry is a fundamental nonequivalence of the arrangements of water molecules in some antipodal configurations with the opposite direction of all H-bonds. For these configurations, the overall pictures of deviations of the hydrogen bonds from linearity are qualitatively different. We consider the reversal of all H-bonds as an additional nongeometric operation of symmetry, more precisely antisymmetry. It is not easy to find the explicit breaking of the symmetry of hydrogen bonding (H-symmetry) in the variety of all configurations. Therefore, this asymmetry may be named hidden.