  An important characteristic of the Dilaton cosmological model is the
Gasperini-Veneziano duality transformation which follows from the existence of
the $O\left( d,d\right) $ symmetry. In this study, we consider the equivalent
Dilaton theory in teleparallel dark energy with the $O\left( d,d\right) $
symmetry, while the equivalent teleparallel-duality transformation is
presented. The classical solution of the field equations is derived. Finally
the Wheeler-DeWitt equation of quantum cosmology is discussed.
