  Let $K$ denote a field and let $V$ denote a vector space over $K$ with finite
positive dimension. We consider an ordered 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 Leonard pair on $V$. Let $x$, $c$, $x^*$, $c^*$ denote
scalars in $K$ with $x$, $x^*$ nonzero, and note that $xA+cI$, $x^*A^* + c^*I$
is a Leonard pair on $V$. We give necessary and sufficient conditions for this
Leonard pair to be isomorphic to the Leonard pair $A$, $A^*$. We also give
necessary and sufficient conditions for this Leonard pair to be isomorphic to
the Leonard pair $A^*$, $A$.
