  In this study, we thoroughly investigate the covariant and $B$-field gauge
invariant odd-parity NS-NS couplings at order $\alpha'^3$, while considering
the removal of field redefinitions, Bianchi identities, and total derivative
freedoms. Our comprehensive analysis reveals the existence of 477 independent
couplings. To establish a specific basis, we construct it in such a way that
none of the couplings contain terms involving structures such as $R$,
$R_{\mu\nu}$, $\nabla_\mu H^{\mu\alpha\beta}$, $\nabla_\mu\nabla^\mu\Phi$, or
terms with more than two derivatives, except for one term that possesses three
derivatives on $H$. Interestingly, the mentioned coupling with the
four-derivative on the $B$-field is rendered zero by the sphere-level
three-point S-matrix element.
  Furthermore, we demonstrate that the remaining 476 parameters in type II
superstring theory are fixed to zero by imposing the requirement that the
circular reduction of the couplings remains invariant under $O(1,1,\MZ)$
T-duality transformations. This result is consistent with our expectations and
highlights the crucial role played by the $O(1,1,\MZ)$ symmetry in constraining
the parameter space of the classical effective actions in string theory.
