  Let $V$ denote a vector space with finite positive dimension. We consider a
pair of linear transformations $A : V \to V$ and $A^* : V \to V$ that satisfy
(i) and (ii) below:
  (i) There exists a basis for $V$ with respect to which the matrix
representing $A$ is irreducible tridiagonal and the matrix representing $A^*$
is diagonal.
  (ii) There exists a basis for $V$ with respect to which the matrix
representing $A^*$ is irreducible tridiagonal and the matrix representing $A$
is diagonal.
  We call such a pair a {\em Leonard pair} on $V$. Let $v_0,v_1,...,v_d$ (resp.
$w_0,w_1,...,w_d$) denote a basis for $V$ referred to in (i) (resp. (ii)). We
show that there exists a unique linear transformation $S: V \to V$ that sends
$v_0$ to a scalar multiple of $v_d$, fixes $w_0$, and sends $w_i$ to a scalar
multiple of $w_i$ for $1 \leq i \leq d$. We call $S$ the {\it switching
element}. We describe $S$ from many points of view.
