We construct a commutative algebra
A
z
\mathcal A_{z}
, generated by
d
d
algebraically independent
q
q
-difference operators acting on variables
z
1
,
z
2
,
…
,
z
d
z_1,z_2,\dots ,z_d
, which is diagonalized by the multivariable Askey-Wilson polynomials
P
n
(
z
)
P_n(z)
considered by Gasper and Rahman (2005). Iterating Sears’
4
ϕ
3
{}_4\phi _3
transformation formula, we show that the polynomials
P
n
(
z
)
P_n(z)
possess a certain duality between
z
z
and
n
n
. Analytic continuation allows us to obtain another commutative algebra
A
n
\mathcal A_{n}
, generated by
d
d
algebraically independent difference operators acting on the discrete variables
n
1
,
n
2
,
…
,
n
d
n_1,n_2,\dots ,n_d
, which is also diagonalized by
P
n
(
z
)
P_n(z)
. This leads to a multivariable
q
q
-Askey-scheme of bispectral orthogonal polynomials which parallels the theory of symmetric functions.