Ice exhibits exceptional structural diversity. This study presents experimental evidence for ice XIX, a second partially-ordered hydrogen phase of ice VI, challenging the assumption of a one-to-one correspondence between disordered and ordered ice phases. Dielectric and neutron diffraction measurements under high pressure demonstrate that disordered ice VI undergoes different hydrogen ordering pathways, thermodynamically controlled by pressure. This multiplicity, potentially present in all disordered ices, provides insights into ice's centrosymmetry and potential (anti-)ferroelectricity, advancing our understanding of ice's phase diagram.