Understanding the structural tendencies of nanoconfined water is of great interest for nanoscience and biology, where nano/micro-sized objects may be separated by very few layers of water. Here we investigate the properties of ice confined to a quasi-2D monolayer by a featureless, chemically neutral potential, in order to characterize its intrinsic behaviour. We use density-functional theory simulations with a non-local van der Waals density functional. An ab initio random structure search reveals all the energetically competitive monolayer configurations to belong to only two of the previously-identified families, characterized by a square or honeycomb hydrogen-bonding network, respectively. We discuss the modified ice rules needed for each network, and propose a simple point dipole 2D lattice model that successfully explains the energetics of the square configurations. All identified stable phases for both networks are found to be non-polar (but with a topologically non-trivial texture for the square) and, hence, non-ferroelectric, in contrast to previous predictions from a five-site empirical force-field model. Our results are in good agreement with very recently reported experimental observations.