The low-temperature phase behavior of liquid water remains unknown. Several scenarios have been proposed, but cannot be distinguished by current experimental data. Here we combine mean field calculations and Monte Carlo simulations of a microscopic cell model. We show that the key physical quantities that establish which of the four proposed scenarios best describes water are two parameters: the strength of the hydrogen bond cooperativity and the strength of the hydrogen bond interaction. We find that all four scenarios are specific cases of a more general scheme which may be mapped in the space of these two parameters. We further show that the results from a thermodynamic free-energy model are consistent with our analysis, suggesting that our conclusion may be model-independent.