A molecular model of network-forming liquids has been formulated in terms of a lattice fluid in which bond formation depends strongly on molecular orientations and local density. The model has been solved in the zeroth- and first-order approximations for molecular and bond geometries similar to water or silica. Results are presented in the form of fluid-phase boundaries, limits of stability, and loci of density extrema. At low temperatures and high pressures the first-order solution shows a liquid-liquid transition with upper and lower consolute temperatures, in addition to vapor-liquid equilibrium; and two loci of density extrema. In contrast to previous models which display a second fluid-fluid transition, the phase behavior of the present model fluid does not result from an a priori imposed dependence of the bonding interaction upon bulk density. The zeroth-order solution shows only one, vapor-liquid, phase transition; and a single, continuous locus of density maxima. The results suggest that low-temperature polyamorphic phase transitions in a pure substance can arise from orientation-dependent interactions; and in particular that a phase transition between two dense fluids can be driven by the greater orientational entropy of high density states which are not fully bonded. The predicted phase behavior is, however, sensitive to the level of approximation used to solve the partition function.