  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 the following four conditions: (i) Each of
A,A* is diagonalizable; (ii) there exists an ordering {V_i}_{i=0}^d of the
eigenspaces of A such that A*V_i\subseteq V_{i-1}+V_i+V_{i+1} for 0\leq i\leq
d, where V_{-1}=0 and V_{d+1}=0; (iii) there exists an ordering
{V*_i}_{i=0}^{\delta} of the eigenspaces of A* such that AV*_i\subseteq
V*_{i-1}+V*_i+V*_{i+1} for 0\leq i\leq\delta, where V*_{-1}=0 and
V*_{\delta+1}=0; (iv) there does not exist a subspace W of V such that
AW\subseteq W, A*W\subseteq W, W\neq0, W\neq V. We call such a pair a TD pair
on V. It is known that d=\delta; to avoid trivialities assume d\geq 1. We show
that there exists a unique linear transformation \Delta:V\to V such that
(\Delta -I)V*_i\subseteq V*_0+V*_1+...+V*_{i-1} and
\Delta(V_i+V_{i+1}+...+V_d)=V_0 +V_{1}+...+V_{d-i} for 0\leq i \leq d. We show
that there exists a unique linear transformation \Psi:V\to V such that \Psi
V_i\subseteq V_{i-1}+V_i+V_{i+1} and (\Psi-\Lambda)V*_i\subseteq
V*_0+V*_1+...+V*_{i-2} for 0\leq i\leq d, where
\Lambda=(\Delta-I)(\theta_0-\theta_d)^{-1} and \theta_0 (resp \theta_d) denotes
the eigenvalue of A associated with V_0 (resp V_d). We characterize \Delta,\Psi
in several ways. There are two well-known decompositions of V called the first
and second split decomposition. We discuss how \Delta,\Psi act on these
decompositions. We also show how \Delta,\Psi relate to each other. Along this
line we have two main results. Our first main result is that \Delta,\Psi
commute. In the literature on TD pairs there is a scalar \beta used to describe
the eigenvalues. Our second main result is that each of \Delta^{\pm 1} is a
polynomial of degree d in \Psi, under a minor assumption on \beta.
