  The Askey-Wilson polynomials are orthogonal polynomials in $x = \cos \theta$,
which are given as a terminating $_4\phi_3$ basic hypergeometric series. The
non-symmetric Askey-Wilson polynomials are Laurent polynomials in
$z=e^{i\theta}$, which are given as a sum of two terminating $_4\phi_3$'s. They
satisfy a biorthogonality relation. In this paper new orthogonality relations
for single $_4\phi_3$'s which are Laurent polynomials in $z$ are given, which
imply the non-symmetric Askey-Wilson biorthogonality. These results include
discrete orthogonality relations. They can be considered as a classical
analytic study of the results for non-symmetric Askey-Wilson polynomials which
were previously obtained by affine Hecke algebra techniques.
