Ice XIX represents the latest discovery of ice polymorphs and exists in the medium pressure range near 1-2 GPa. Ice XIX is a partially hydrogen-ordered phase, by contrast to its disordered mother phase ice VI, which shares the same oxygen-atom network with ice XIX. Ice XIX differs in terms of the ordering of the hydrogen-atom sublattice, and hence the space group, from its hydrogen-ordered sibling ice XV, which also features the same type of oxygen network. Together, ice VI, XV, and XIX form the only known trio of ice polymorphs, where polymorphic transformations from order to order, order to disorder, and disorder to order are possible, which also compete with each other depending on the thermodynamic path taken and the cooling/heating rates employed. These transitions in the H-sublattice have barely been investigated, so we study here the unique triangular relation in the ice VI/XV/XIX trio based on calorimetry experiments. We reveal the following key features for H-sublattice transitions: (i) upon cooling ice VI, domains of ice XV and XIX develop simultaneously, where pure ice XV forms at ≤0.85 GPa and pure ice XIX forms at ≥1.60 GPa, (ii) ice XIX transforms into ice XV via a transient disordered state, (iii) ice XV recooled at ambient pressure features a complex domain structure, possibly containing an unknown H-ordered polymorph, (iv) recooled ice XV partly transforms back into ice XIX at 1.80 GPa, and (v) partial deuteration slows down domain reordering strongly. These findings not only are of interest in understanding possible hydrogen-ordering and -disordering processes in the interior of icy moons and planets but, more importantly, also provide a challenging benchmark for our understanding and parameterizing many-body interactions in H-bonded networks.