Several of the high-pressure crystalline forms of ice exist in the interiors of the larger icy moons of the outer planets. Guided by the phase relations established by Bridgman 75 years ago, we have transformed ordinary water ice I to ices II, III, V, and perhaps VI. We subsequently deformed those phases in a gas deformation apparatus designed for these experiments. We found ice II to be the strongest of the high-pressure phases, having a strength roughly comparable to that of ice I. Ice II is very weak: at fixed levels of stress, ice III flows 102--103 times faster than ice II. We confirm Bridgman's (1912) findings that ices III and V can exist metastably within the ice II field and, in addition, find that these phases may be deformed plastically within much of the metastable region without reverting to ice II. The most persistent metastable existence of ice III occurs in the warmer portions of the ice II field, through which historical and present-day temperature-pressure profiles in the icy moons pass. Thus, although the stability field of ice III is small, the weakness of this phase may have profoundly influenced the evolution and present-day behavior of the icy moons.