Spontaneous self-assembly of hard convex polyhedra is known to form orientationally disordered crystalline phases, where particle orientations do not follow the same pattern as the positional arrangement of the crystal. A distinct type of orientational phase with discrete rotational mobility has been reported in hard particle systems. In this paper, we present a new analysis method for characterizing the orientational phase of a crystal, which is based on algorithmic detection of unique orientations. Using this method, we collected complete statistics of discrete orientations along the Monte Carlo simulation trajectories and observed that particles were equally partitioned among them, with specific values of pairwise orientational differences. These features remained constant across the pressure range and did not depend on rotational mobility. The discrete mobility was characteristic of a distinct equilibrium thermodynamic phase, qualitatively different from the freely rotating plastic phase with continuous orientations. The high pressure behavior with frozen particle orientations was part of that same description and not a non-equilibrium arrested state. We introduced a precise notion of orientational order and demonstrated that the system was maximally disordered at the level of a unit cell, even though individual particles could only take a few discrete orientations. We report the existence of this phase in five polyhedral shapes and in systematically curated shape families constructed around two of them. The symmetry mismatch between the particle and the crystallographic point groups was found to be a predictive indicator for the occurrence of this phase.
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