It has recently been shown that the TIP4P/Ice model of water can be studied numerically in metastable equilibrium at and below its liquid-liquid critical temperature. We report here simulations along a subcritical isotherm, for which two liquid states with the same pressure and temperature but different density can be equilibrated. This allows for a clear visualization of the structural changes taking place across the transition. We specifically focus on how the topological properties of the H-bond network change across the liquid-liquid transition. Our results demonstrate that the structure of the high-density liquid, characterized by the existence of interstitial molecules and commonly explained in terms of the collapse of the second neighbor shell, actually originates from the folding back of long rings, bringing pairs of molecules separated by several hydrogen-bonds close by in space.