In this work we study through computer simulations the three known forms of amorphous ice, namely, the low density (LDA), the high density (HDA), and the very high density (VHDA) amorphous ices, at various pressures and temperatures. Adopting the TIP4P model, we are able to reproduce these three forms by following experiment-like procedures. Those systems are characterized from thermodynamic and structural points of view, in particular through an insightful analysis of the behavior of the second-shell neighbors in the various stages of the simulations. The distance-ranked neighbors are found to be the relevant markers of the differentiation mechanisms of the three forms, since their response to pressure induces specific distortions of the orientational correlations. We show that LDA, HDA, and VHDA are disordered forms whose local structures tend to a single tetrahedral network, to an ice-VII-like arrangement, and to a random-close-packed simple liquidlike structure, respectively. Despite the major structural deformations induced by pressure, the hydrogen bonds are still present in each of those forms, even though deformations of the tetrahedral angles are necessary to adapt to the denser structures.