Geometric permutations of disjoint unit spheres

We show that a set of n disjoint unit spheres in Rd admits at most two distinct geometric permutations if n⩾9, and at most three if 3⩽n⩽8. This result improves a Helly-type theorem on line transversals for disjoint unit spheres in R3: if any subset of size at most 18 of a family of such spheres admits a line transversal, then there is a line transversal for the entire family.