  We observe that conformal blocks of scalar 4-point functions in a
$d$-dimensional conformal field theory can mapped to eigenfunctions of a
2-particle hyperbolic Calogero-Sutherland Hamiltonian. The latter describes two
coupled P\"oschl-Teller particles. Their interaction, whose strength depends
smoothly on the dimension $d$, is known to be superintegrable. Our observation
enables us to exploit the rich mathematical literature on Calogero-Sutherland
models in deriving various results for conformal field theory. These include an
explicit construction of conformal blocks in terms of Heckman-Opdam
hypergeometric functions and a remarkable duality that relates the blocks of
theories in different dimensions.
