  A square matrix is called {\it Hessenberg} whenever each entry below the
subdiagonal is zero and each entry on the subdiagonal is nonzero. Let $V$
denote a nonzero finite-dimensional vector space over a field $\fld$. We
consider an ordered pair of linear transformations $A: V \to V$ and $A^*: V \to
V$ which satisfy both (i), (ii) below. \begin{enumerate} \item There exists a
basis for $V$ with respect to which the matrix representing $A$ is Hessenberg
and the matrix representing $A^*$ is diagonal.
  \item There exists a basis for $V$ with respect to which the matrix
representing $A$ is diagonal and the matrix representing $A^*$ is Hessenberg.
\end{enumerate}
  \noindent We call such a pair a {\it thin Hessenberg pair} (or {\it TH
pair}). This is a special case of a {\it Hessenberg pair} which was introduced
by the author in an earlier paper. We investigate several bases for $V$ with
respect to which the matrices representing $A$ and $A^*$ are attractive. We
display these matrices along with the transition matrices relating the bases.
We introduce an "oriented" version of $A,A^*$ called a TH system. We classify
the TH systems up to isomorphism.
