  We discuss the relation between Noether (point) symmetries and discrete
symmetries for a class of minisuperspace cosmological models. We show that when
a Noether symmetry exists for the gravitational Lagrangian then there exists a
coordinate system in which a reversal symmetry exists. Moreover as far as
concerns the scale-factor duality symmetry of the dilaton field, we show that
it is related to the existence of a Noether symmetry for the field equations,
and the reversal symmetry in the normal coordinates of the symmetry vector
becomes scale-factor duality symmetry in the original coordinates. In
particular the same point symmetry as also the same reversal symmetry exists
for the Brans-Dicke- scalar field with linear potential while now the discrete
symmetry in the original coordinates of the system depends on the Brans-Dicke
parameter and it is a scale-factor duality when $\omega_{BD}=1$. Furthermore,
in the context of the O'Hanlon theory for $f\left( R\right) $-gravity, it is
possible to show how a duality transformation in the minisuperspace can be used
to relate different gravitational models.
