The many different intermolecular bonding interactions present in seven of the phases of water ice representing a large portion of the phase diagram are systematically examined using the theory of "atoms in molecules" (AIM), we obtain the total charge density distributions calculated using the generalized gradient approximation (GGA) and norm-conserving pseudopotentials. With this approach, we follow the changing properties of the intermolecular bonding from the low- to high-pressure phases of ice, examining ice Ic, ice Ih, ice IX, ice II, ice VI, ice VII, and ice Vm and various proton orderings within these phases. We quantify and characterize not only the familiar hydrogen bonds but also some novel intermolecular O ... O bonding interactions, enhancing the understanding of bonding in ice and allowing a more complete explanation of the unusually high number of distinct phases of ice. We also quantify the relative compressibility of the structural features and the bond path rigidity in these ice phases, which have consequences for the mechanisms involved in phase transformations.