Ice is a material of fundamental importance for a wide range of scientific disciplines including physics, chemistry, and biology, as well as space and materials science. A well-known feature of its phase diagram is that high-temperature phases of ice with orientational disorder of the hydrogen-bonded water molecules undergo phase transitions to their ordered counterparts upon cooling. Here, we present an example where this trend is broken. Instead, hydrochloric-acid-doped ice VI undergoes an alternative type of phase transition upon cooling at high pressure as the orientationally disordered ice remains disordered but undergoes structural distortions. As seen with in-situ neutron diffraction, the resulting phase of ice, ice XIX, forms through a Pbcn-type distortion which includes the tilting and squishing of hexameric clusters. This type of phase transition may provide an explanation for previously observed ferroelectric signatures in dielectric spectroscopy of ice VI and could be relevant for other icy materials.