Liquid water is considered poorly understood. How are water's physical properties encoded in its molecular structure? We introduce a statistical mechanical model (CageWater) of water's hydrogen-bonding (HB) and Lennard-Jones (LJ) interactions. It predicts the energetic and volumetric and anomalous properties accurately. Yet, because the model is analytical, it is essentially instantaneous to compute. This model advances our understanding beyond current molecular simulations and experiments. Water has long been regarded as a "2-density liquid": a dense LJ liquid and a looser HB one. Instead, we find here a different antagonism underlying water structure-property relations: HBs in water-water pairs drive density, while HBs in cooperative cages drive openness. The balance shifts strongly with temperature and pressure. This model interprets the molecular structures underlying the liquid liquid phase transition in supercooled water. It may have value in geophysics, biomolecular modeling, and engineering of materials for water purification and green chemistry.