Each unit-quaternion, representing the rotation of a given molecule in an arbitrary crystal dataset, may exist anywhere on the S 3 hyper-sphere. Thus its four marginal coordinates [q 0 , q 1 , q 2 , q 3 ] are coupled by the unit-length constraint. Moreover, antipodal quaternions represent equivalent rotation matrices (i.e., R(q) = R(-q)). This means that any hemisphere of S 3 is sufficient to describe all possible rotations, or equivalently, only half of S 3 is sufficient to specify a set of all unique quaternion-based representations of all 3D rotations. When considering the curvature of S 3 , or its hemisphere, it is clear that an explicit global parametrization of this space is impossible without encountering singularities. Instead, the recommended singularityfree approach is to work directly with unit vectors embedded in R 4 (implicit representation). Indeed, this is exactly how Ref. approached the problem of mapping between distributions of rotations. Specifically, two novel flip-equivariant bijectors (symmetrized Moebius transformations and symmetrized projective convex gradient maps) were developed in Ref. that enable modeling smooth flip-symmetric flows on the surface of any flip-symmetric sphere.