  We derive a set of no-go theorems and yes-go examples for the parity-odd
primordial trispectrum of curvature perturbations. We work at tree-level in the
decoupling limit of the Effective Field Theory of Inflation and assume scale
invariance and a Bunch-Davies vacuum. We show that the parity-odd scalar
trispectrum vanishes in the presence of any number of scalar fields with
arbitrary mass and any parity-odd scalar correlator vanishes in the presence of
any number of spinning fields with massless de Sitter mode functions, in
agreement with the findings of Liu, Tong, Wang and Xianyu [1]. The same is true
for correlators with an odd number of conformally-coupled external fields. We
derive these results using both the (boostless) cosmological bootstrap, in
particular the Cosmological Optical Theorem, and explicit perturbative
calculations. We then discuss a series of yes-go examples by relaxing the above
assumptions one at the time. In particular, we provide explicit results for the
parity-odd trispectrum for (i) violations of scale invariance in single-clock
inflation, (ii) the modified dispersion relation of the ghost condensate
(non-Bunch-Davies vacuum), and (iii) interactions with massive spinning fields.
Our results establish the parity-odd trispectrum as an exceptionally sensitive
probe of new physics beyond vanilla inflation.
