isomorphic to ${{\Bbb Z}}_2\times{{\Bbb Z}}_2$. The group action is without fixed points, since the eigenvalues $\theta_i$’s (or $\theta_i^*$’s) are distinct. Let $\downarrow$ and $\Downarrow$ denote the permutations which transform (\[parray1\]) into, respectively, (\[parray2\]) and (\[parray3\]). Observe that the composition $\downarrow\Downarrow$ transforms (\[parray1\]) into (\[parray9\]). We refer to the permutations $\downarrow$, $\Downarrow$ and $\downarrow\Downarrow$ as [*relation operators*]{}, because in [@terwgen Section 4] the parameter arrays in (\[parray1\])–(\[parray9\]) corresponding to $(A,A^*)$ and the 4 similar parameter arrays corresponding to the Leonard pair $(A^*,A)$ are called [*relatives*]{} of each other.

Parameter arrays are classified by Terwilliger in [@TerwLTparr]; alternatively, see [@terwgen Section 35]. For each parameter array, certain orthogonal polynomials naturally occur in entries of the transformation matrix between two bases characterized in Definition \[deflp\] for the corresponding Leonard pair. Terwilliger’s classification largely mimics the terminating branch of orthogonal polynomials in the Askey-Wilson scheme [@koekswart]. Specifically, the classification comprises Racah, Hahn, Krawtchouk polynomials and their $q$-versions, plus Bannai-Ito and orphan polynomials. Classes of parameter arrays can be identified by the type of corresponding orthogonal polynomials; we refer to them as [*Askey-Wilson types*]{}. The type of a parameter array is unambiguously defined if $d\ge 3$. We recapitulate Terwilliger’s classification in Sections \[qpararrays1\] through \[otherpararrays\] 