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We employ classical molecular dynamics simulations to investigate the molecular-level structure of water during the isothermal compression of hexagonal ice (Ih) and low-density amorphous (LDA) ice at low temperatures. In both cases, the system transforms to high-density amorphous ice (HDA) via a first-order-like phase transition. We employ a sensitive local order metric (LOM) [F. Martelli et al., Phys. Rev. B 97, 064105 (2018)] that can discriminate among different crystalline and noncrystalline ice structures and is based on the positions of the oxygen atoms in the first- and/or second-hydration shell. Our results confirm that LDA and HDA are indeed amorphous, i.e., they lack polydispersed ice domains. Interestingly,HDAcontains a small number of domains that are reminiscent of the unit cell of ice IV, although the hydrogen-bond network (HBN) of these domains differs from the HBN of ice IV. The presence of ice-IV-like domains provides some support to the hypothesis that HDA could be the result of a detour on the HBN rearrangement along the Ih-to-ice-IV pressure-induced transformation. Both nonequilibrium LDA-to-HDA and Ih-to-HDA transformations are two-step processes where a small distortion of the HBN first occurs at low pressures and then, a sudden, extensive rearrangement of hydrogen bonds at the corresponding transformation pressure follows. Interestingly, the Ih-to-HDA and LDA-to-HDA transformations occur when LDA and Ih have similar local order, as quantified by the site-averaged LOMs. Since Ih has a perfect tetrahedral HBN while LDA does not, it follows that higher pressures are needed to transform Ih into HDA than that for the conversion of LDAtoHDA. In correspondencewith both first-order-like phase transitions, the samples are composed of a large HDA cluster that percolates within the Ih/LDA samples. Our results shed light on the debated structural properties of amorphous ices and indicate that the kinetics of the Ih-to-HDA and LDA-to-HDA transformations require an in-depth inspection of the underlying HBN. Such investigation is currently ongoing.