  We show that dynamical Chern-Simons (dCS) gravity imprints a parity-violating
signal in primordial scalar perturbations. Specifically, we find that, after
dCS amplifies one graviton helicity due to a tachyonic instability, the
graviton-mediated correlation between two pairs of scalars develops a
parity-odd component. This correlation, the primordial scalar trispectrum, is
then transferred to the corresponding curvature correlator and thus is
imprinted in both LSS and the CMB. We find that the parity-odd piece has
roughly the same amplitude as its parity-even counterpart, scaled linearly by
the degree of gravitational circular polarization $\Pi_{\rm circ} \sim
\sqrt{\varepsilon}[H^2/(M_{\rm Pl} f)] \leq 1$, with $\varepsilon$ the
slow-roll parameter, $H$ the inflationary Hubble scale, $f$ the dCS decay
constant, and the upper bound saturated for purely circularly-polarized
gravitons. We also find that, in the collapsed limit, the ratio of the two
trispectra contains direct information about the graviton's spin. In models
beyond standard inflationary dCS, e.g. those with multiple scalar fields or
superluminal scalar sound speed, there can be a large enhancement factor $F
\gtrsim 10^6$ to the trispectrum. We find that an LSS survey that contains
$N_{\rm modes}$ linear modes would place an $n\sigma$ constraint on $\Pi_{\rm
circ}r$ of $\sim 0.04\ (n/3)(10^6/F)(10^6/N_{\rm modes})^{1/2}$ from the
parity-odd galaxy trispectrum, for tensor-to-scalar ratio $r$. We also forecast
for several spectroscopic and 21-cm surveys. This constraint implies that, for
high-scale single-field inflation parameters, LSS can probe very large dCS
decay constants $f \lesssim 4\times 10^9\ {\rm GeV}(3/n)(F/10^6)\left(N_{\rm
modes}/10^6\right)^{1/2}$. Our result is the first example of a massless
particle yielding a parity-odd scalar trispectrum through spin-exchange.
