  Understanding how particles are arranged on the sphere is not only central to
numerous physical, biological, and materials systems but also finds
applications in mathematics and in analysis of geophysical and meteorological
measurements. In contrast to particle distributions in Euclidean space,
restriction that the particles should lie on the sphere brings about several
important constraints. These require a careful extension of quantities used for
particle distributions in Euclidean space to those confined to the sphere. We
introduce a framework designed to analyze and classify structural (dis)order in
particle distributions constrained to the sphere. The classification is based
on the concept of hyperuniformity, which was introduced 15 years ago and since
then studied extensively in Euclidean space, yet has only very recently been
considered for the sphere. We build our framework on a generalization of the
structure factor on the sphere, which we relate to the power spectrum of the
corresponding multipole expansion. The spherical structure factor is then shown
to couple with cap number variance, a measure of local density fluctuations,
allowing us to derive different forms of the variance. In this way, we
construct a classification of hyperuniformity for scale-free particle
distributions on the sphere and show how it can be extended to other
distributions as well. We demonstrate that hyperuniformity on the sphere can be
defined either through a vanishing spherical structure factor at low multipole
numbers or through a scaling of the cap number variance, in both cases
extending the Euclidean definition while pointing out crucial differences. Our
work provides a comprehensive tool for detecting long-range order on spheres
and the analysis of spherical computational meshes, biological and synthetic
spherical assemblies, and ordering phase transitions in spherically-distributed
particles.
