Ab initio molecular dynamics simulations of liquid water under equilibrium ambient conditions, together with a novel energy decomposition analysis, have recently shown that a substantial fraction of water molecules exhibit a significant asymmetry between the strengths of the two donor and/or the two acceptor interactions. We refer to this recently unraveled aspect as the "local asymmetry in the hydrogen bond network". We discuss how this novel aspect was first revealed, and provide metrics that can be consistently employed on simulated water trajectories to quantify this local heterogeneity in the hydrogen bond network and its dynamics. We then discuss the static aspects of the asymmetry, pertaining to the frozen geometry of liquid water at any given instant of time and the distribution of hydrogen bond strengths therein, and also its dynamic characteristics pertaining to how fast this asymmetry decays and the kinds of molecular motions responsible for this decay. Following this we discuss the spectroscopic manifestations of this asymmetry, from ultrafast X-ray absorption spectra to infrared spectroscopy and down to the much slower terahertz regime. Finally, we discuss the implications of these findings in a broad context and their relation to the current notions about the structure and dynamics of liquid water.