  How can detector click probabilities respond to spatial rotations around a
fixed axis, in any possible physical theory? Here, we give a thorough
mathematical analysis of this question in terms of "rotation boxes", which are
analogous to the well-known notion of non-local boxes. We prove that quantum
theory admits the most general rotational correlations for spins 0, 1/2, and 1,
but we describe a metrological game where beyond-quantum resources of spin 3/2
outperform all quantum resources of the same spin. We prove a multitude of
fundamental results about these correlations, including an exact convex
characterization of the spin-1 correlations, a Tsirelson-type inequality for
spins 3/2 and higher, and a proof that the general spin-J correlations provide
an efficient outer SDP approximation to the quantum set. Furthermore, we review
and consolidate earlier results that hint at a wealth of applications of this
formalism: a theory-agnostic semi-device-independent randomness generator, an
exact characterization of the quantum (2,2,2)-Bell correlations in terms of
local symmetries, and the derivation of multipartite Bell witnesses. Our
results illuminate the foundational question of how space constrains the
structure of quantum theory, they build a bridge between
semi-device-independent quantum information and spacetime physics, and they
demonstrate interesting relations to topics such as entanglement witnesses,
spectrahedra, and orbitopes.
