  Let $M$ be a connected compact orientable surface, $f:M\to \mathbb{R}$ be a
Morse function, and $h:M\to M$ be a diffeomorphism which preserves $f$ in the
sense that $f\circ h = f$. We will show that if $h$ leaves invariant each
regular component of each level set of $f$ and reverses its orientation, then
$h^2$ is isotopic to the identity map of $M$ via $f$-preserving isotopy.
  This statement can be regarded as a foliated and a homotopy analogue of a
well known observation that every reversing orientation orthogonal isomorphism
of a plane has order $2$, i.e. is a mirror symmetry with respect to some line.
The obtained results hold in fact for a larger class of maps with isolated
singularities from connected compact orientable surfaces to the real line and
the circle.
