We construct a commutative algebra A_z, generated by d algebraically
independent q-difference operators acting on variables z_1, z_2,..., z_d, which
is diagonalized by the multivariable Askey-Wilson polynomials P_n(z) considered
by Gasper and Rahman [6]. Iterating Sears' transformation formula, we show that
the polynomials P_n(z) possess a certain duality between z and n. Analytic
continuation allows us to obtain another commutative algebra A_n, generated by
d algebraically independent difference operators acting on the discrete
variables n_1, n_2,..., n_d, which is also diagonalized by P_n(z). This leads
to a multivariable q-Askey-scheme of bispectral orthogonal polynomials which
parallels the theory of symmetric functions.