  We study the parity-breaking higher-curvature gravity theory of Chern-Simons
(CS), using the Palatini formulation in which the metric and connection are
taken to be independent fields. We first show that Palatini CS gravity leads to
first-order derivative equations of motion and thus avoid the typical
instabilities of CS gravity in the metric formalism. As an initial application,
we analyze the cosmological propagation of gravitational waves (GWs) in
Palatini CS gravity. We show that, due to parity breaking, the polarizations of
GWs suffer two effects during propagation: amplitude birefringence (which
changes the polarization ellipticity) and velocity birefringence (which rotates
the polarization plane). While amplitude birefringence is known to be present
in CS gravity in the metric formalism, velocity birefringence is not present in
metric CS gravity, but now appears in Palatini CS due to the fact that
left-handed and right-handed GW polarizations have a different dispersion
relation. In the approximation of small deviations from General Relativity
(GR), we do find however that velocity birefringence appears at least
quadratically in the CS coupling parameter $\alpha$, while amplitude
birefringence appears linearly in $\alpha$. This means that amplitude
birefringence will be the most relevant effect in Palatini CS and hence this
model will behave similarly to metric CS. We confirm this by applying current
constraints on amplitude and velocity birefringence to Palatini CS, and showing
that those from amplitude birefringence give the tightest bounds.
