The interactions of a hydrophilic surface with water can significantly influence the characteristics of the liquid water interface. In this manuscript, we explore this influence by studying the molecular structure of liquid water at a disordered surface with tunable surfacewater interactions. We combine all-atom molecular dynamics simulations with a mean field model of interfacial hydrogen bonding to analyze the effect of surfacewater interactions on the structural and energetic properties of the liquid water interface. We find that the molecular structure of water at a weakly interacting (i.e., hydrophobic) surface is resistant to change unless the strength of surfacewater interactions is above a certain threshold. We find that below this threshold water's interfacial structure is homogeneous and insensitive to the details of the disordered surface, however, above this threshold water's interfacial structure is heterogeneous. Despite this heterogeneity, we demonstrate that the equilibnum distribution of molecular orientations can be used to quantify the energetic component of the surfacewater interactions that contribute specifically to modifying the interfacial hydrogen bonding network. We identify this specific energetic component as a new measure of hydrophilicity, which we refer to as the intrinsic hydropathy.