  Conformal blocks for correlation functions of tensor operators play an
increasingly important role for the conformal bootstrap programme. We develop a
universal approach to such spinning blocks through the harmonic analysis of
certain bundles over a coset of the conformal group. The resulting Casimir
equations are given by a matrix version of the Calogero-Sutherland Hamiltonian
that describes the scattering of interacting spinning particles in a
1-dimensional external potential. The approach is illustrated in several
examples including fermionic seed blocks in 3D CFT where they take a very
simple form.
