  The connection between the recoupling scheme of four copies of
$\mathfrak{su}(1,1)$, the generic superintegrable system on the 3 sphere, and
bivariate Racah polynomials is identified. The Racah polynomials are presented
as connection coefficients between eigenfunctions separated in different
spherical coordinate systems and equivalently as different irreducible
decompositions of the tensor product representations. As a consequence of the
model, an extension of the quadratic algebra ${\rm QR}(3)$ is given. It is
shown that this algebra closes only with the inclusion of an additional shift
operator, beyond the eigenvalue operators for the bivariate Racah polynomials,
whose polynomial eigenfunctions are determined. The duality between the
variables and the degrees, and hence the bispectrality of the polynomials, is
interpreted in terms of expansion coefficients of the separated solutions.
