  Given the growing interest in gravitational-wave and cosmological
parity-violating effects in dynamical Chern-Simons (dCS) gravity, it is crucial
to investigate whether the scalar-gravitational Pontryagin term in dCS persists
when formulated in the context of the $\text{U(1)}_{\text{B}-\text{L}}$ anomaly
in the Standard Model (SM). In particular, it has been argued that dCS gravity
can be reduced to Einstein gravity after ''rotating away'' the
gravitational-Pontryagin coupling into the phase of the Weinberg operator
$\unicode{x2013}$ analogous to the rotation of the axion zero-mode into the
quark mass matrix. We find that dCS is nontrivial if the scalar field $\phi$
has significant space-time dependence from dynamics. We provide a comprehensive
consideration of the dCS classical and quantum symmetries relevant for
embedding a dCS sector in the SM. We find that, because of the B-L chiral
gravitational anomaly, the scalar-Pontryagin term cannot be absorbed by a field
redefinition. Assuming a minimal extension of the SM, we also find that a
coupling of the dCS scalar with right-handed neutrinos induces both the
scalar-Pontryagin coupling and an axion-like phase in the dimension-five
Weinberg operator. We comment on the issue of gauging the
$\text{U(1)}_{\text{B}-\text{L}}$, the observational effects with these two
operators present for upcoming experiments, and the origin of dCS gravity in
string theory.
