  In his famous theorem (1982), Douglas Leonard characterized the $q$-Racah
polynomials and their relatives in the Askey scheme from the duality property
of $Q$-polynomial distance-regular graphs. In this paper we consider a
nonsymmetric (or Laurent) version of the $q$-Racah polynomials in the above
situation. Let $\Gamma$ denote a $Q$-polynomial distance-regular graph that
contains a Delsarte clique $C$. Assume that $\Gamma$ has $q$-Racah type. Fix a
vertex $x \in C$. We partition the vertex set of $\Gamma$ according to the
path-length distance to both $x$ and $C$. The linear span of the characteristic
vectors corresponding to the cells in this partition has an irreducible module
structure for the universal double affine Hecke algebra $\hat{H}_q$ of type
$(C^{\vee}_1, C_1)$. From this module, we naturally obtain a finite sequence of
orthogonal Laurent polynomials. We prove the orthogonality relations for these
polynomials, using the $\hat{H}_q$-module and the theory of Leonard systems.
Changing $\hat{H}_q$ by $\hat{H}_{q^{-1}}$ we show how our Laurent polynomials
are related to the nonsymmetric Askey-Wilson polynomials, and therefore how our
Laurent polynomials can be viewed as nonsymmetric $q$-Racah polynomials.
