The remaining 3n mol DOFs of the system, corresponding to the n mol unit-quaternions (specifying the rotations of the molecule in a supercell relative to a fixed reference frame ensured by the non-rotating supercell vectors), were treated as discussed below. 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 are equivalent in relation to the rotation matrices that they represent (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 quaternionbased representations of all 3D rotations. When considering the curvature of S 3 , or its hemisphere, it is clear that an explicit global parametrisation of this space is impossible without encountering singularities. Instead, the recommended singularity-free 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 modelling smooth flip-symmetric flows on the surface of any flip-symmetric sphere.