We analyze the nature of the structural order established in liquid TIP4P water in the framework provided by the multiparticle correlation expansion of the statistical entropy. Different regimes are mapped onto the phase diagram of the model upon resolving the pair entropy into its translational and orientational components. These parameters are used to quantify the relative amounts of positional and angular order in a given thermodynamic state, thus allowing a structurally unbiased definition of low-density and high-density water. As a result, the structurally anomalous region--within which both types of order are simultaneously disrupted by an increase of pressure at constant temperature--is clearly identified through extensive molecular-dynamics simulations.