Emblematic of polytopal rearrangements, and in many ways the inspiration for this work, is the Berry pseudorotation (BPR) mechanism whereby two trigonal bipyramidal (TBPY-5) configurations interconvert through a square pyramidal (SPY-5) intermediate with an example shown by the green track in Fig. . It is tempting to view the BPR mechanism as a trajectory through the 2-component AB5 TBPY-5 E′-mode configuration space given that molecular modelling presents the doubly degenerate vibrations of 𝑒′ generic symmetry initiate the distortion of a TBPY-5 configuration towards the intermediate SPY-5 configurations. This, however, belies an important technical detail. The subtlety arises by noting that there is no singular generic SPY-5 configuration; there is a continuum of SPY-5 configurations, all with 𝐶 4𝑣 generic symmetry, at the intermediate point for a BPR process with different chemical systems exemplifying different points on this continuum (see Fig. ). Careful examination of the TBPY-5 modes reveals that there are two sets of doubly degenerate E′-modes that account for these different SPY-5 geometries. Consequently, BPR (and its related polytopal rearrangements) traverse an at-least 𝐶 2𝑣 generic symmetry, 2D seam in the four component AB5 TBPY-5 (E′×E′)-mode configuration space (see . A full analysis of this 4D space is well beyond the scope of this paper and will be dealt with elsewhere.