preserve the set of normalized parameter arrays?

  ------------------------------------------------------------ --------------------- ----------------------------------------------------------------
                          Askey-Wilson                            Affine scaling                         Conversion of normalized

                              type                                   $(t,t^*)$                               parameter arrays

                           $q$-Racah                                 $(-1,1)$                          $s\mapsto -s$, $r\mapsto -r$

                                                                     $(1,-1)$                        $s^*\mapsto -s^*$, $r\mapsto -r$

                            $q$-Hahn                             $(\sqrt{-1},-1)$                $s^*\mapsto-s^*$, $r\mapsto\sqrt{-1}\,r$

                         Dual $q$-Hahn                           $(-1,\sqrt{-1})$                  $s\mapsto-s$, $r\mapsto\sqrt{-1}\,r$

                         $q$-Krawtchouk                              $(1,-1)$                               $s^*\mapsto -s^*$

                      Dual $q$-Krawtchouk                            $(-1,1)$                                 $s\mapsto -s$

   $\begin{array}{l} \mbox{Quantum and affine}\vspace{-5pt}\\   $(\zeta_3,\zeta_3)$                        $r\mapsto\zeta_3\,r$
                \mbox{$q$-Krawtchouk}\end{array}$                                    

                             Racah                                       —                                          —

                              Hahn                                   $(-1,1)$                         $\Downarrow$ and $v\mapsto-v$

                           Dual Hahn                                 $(1,-1)$                         $\downarrow$ and $v\mapsto-v$

                           Krawtchouk                                $(-1,1)$                                  $\Downarrow$

                                                                     $(1,-1)$                                  $\downarrow$

                           Bannai-Ito                                $(-1,1)$          If $d$ even: $\Downarrow$ and $u\mapsto-u$, $v\mapsto-v\;\;$

                                                                     $(1,-1)$         If $d$ even: $\downarrow$ and $u^*\!\mapsto-u^*$, $v\mapsto-v$
  ------------------------------------------------------------ --------------------- ----------------------------------------------------------------

  : Reparametrization of different normalizations[]{data-label="normtr"}

  ------------------------------------------------------------ ---------------------------------------------- -------------------------------------------------------
                          Askey-Wilson                                         Change of sign                                     Parameter array

                              type                                        of $\sqrt{\beta\!+\!2}$                                 stays invariant

                           $q$-Racah                                                 —                         $r\mapsto 1/r$; also (\[qracahinv\]), (\[qracahin2\])

                            $q$-Hahn                             $q\mapsto -q$, $s^*\mapsto (-1)^{d+1}s^*$                               —

                         Dual $q$-Hahn                              $q\mapsto-q$, $s\mapsto (-1)^{d+1}s$                                 —

                         $q$-Krawtchouk                         If $d$ odd: $q\mapsto-q$, $s^*\mapsto-s^*\!$                If $d$ even: $q\mapsto -q$

                      Dual $q$-Krawtchouk                       If $d$ odd: $q\mapsto -q$, $s\mapsto-s\;\;$                  If $d$ even: $q\mapsto-q$

   $\begin{array}{l} \mbox{Quantum and affine}\vspace{-5pt}\\       $q\mapsto-q$, $r\mapsto (-1)^{d+1}r$                                 —
                \mbox{$q$-Krawtchouk}\end{array}$                                                             

                             Racah                                                   —                                           $v\mapsto -v-d-1$

                           Bannai-Ito                                                —                                       If $d$ odd: $v\mapsto -v$
  ------------------------------------------------------------ ---------------------------------------------- -------------------------------------------------------

  : Alternative normalization and invariant reparametrizations[]{data-label="invartr"}

Regarding the first question, non-uniqueness occurs for two reasons:

-   There exist affine scalings by 