  We study the curve counting invariants of Calabi--Yau 3-folds via the Weyl
reflection along a ruled divisor. We obtain a new rationality result and
functional equation for the generating functions of Pandharipande--Thomas
invariants. When the divisor arises as resolution of a curve of
$A_1$-singularities, our results match the rationality of the associated
Calabi--Yau orbifold.
  The symmetry on generating functions descends from the action of an infinite
dihedral group of derived auto-equivalences, which is generated by the derived
dual and a spherical twist. Our techniques involve wall-crossing formulas and
generalized DT invariants for surface-like objects.
