  What would a parity-violating universe look like? We present a numerical and
theoretical study of mirror asymmetries in the late universe, using a new suite
of $N$-body simulations: QUIJOTE-Odd. These feature parity-violating initial
conditions, injected via a simple ansatz for the imaginary primordial
trispectrum and evolved into the non-linear regime. We find that the
realization-averaged power spectrum, bispectrum, halo mass function, and matter
PDF are not affected by our modifications to the initial conditions, deep into
the non-linear regime, which we argue arises from rotational and translational
invariance. In contrast, the parity-odd trispectrum of matter (measured using a
new estimator), shows distinct signatures proportional to the parity-violating
parameter, $p_{\rm NL}$, which sets the amplitude of the primordial
trispectrum. We additionally find intriguing signatures in the angular momentum
of halos, with the primordial trispectrum inducing a non-zero correlation
between angular momentum and smoothed velocity field, proportional to $p_{\rm
NL}$. Our simulation suite has been made public to facilitate future analyses.
