Self-dual Leonard pairs

Let $\F$ denote a field and let $V$ denote a vector space over $\F$ with finite positive dimension. Consider a pair $A,A^*$ of diagonalizable $\F$-linear maps on $V$, each of which acts on an eigenbasis for the other one in an irreducible tridiagonal fashion. Such a pair is called a Leonard pair. We consider the self-dual case in which there exists an automorphism of the endomorphism algebra of $V$ that swaps $A$ and $A^*$. Such an automorphism is unique, and called the duality $A \leftrightarrow A^*$. In the present paper we give a comprehensive description of this duality. In particular, we display an invertible $\F$-linear map $T$ on $V$ such that the map $X \mapsto T X T^{-1}$ is the duality $A \leftrightarrow A^*$. We express $T$ as a polynomial in $A$ and $A^*$. We describe how $T$ acts on $4$ flags, $12$ decompositions, and 24 bases for $V$.