  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 $V$.
