  Cliff collapses on Comet 67P/Churyumov-Gerasimenko expose relatively pristine
nucleus matter and offer rare opportunities to characterise ice-rich comet
material. Here, Microwave Instrument for \emph{Rosetta} Orbiter (MIRO)
observations of two collapsed or crumbling cliffs in the Imhotep and Hathor
regions have been assembled. The empirical diurnal antenna temperature curves
are analysed with thermophysical and radiative transfer models in order to
place constraints on the physical properties and degrees of stratification in
the near-surface material. The Imhotep site consists of an exposed
dust/water-ice mixture with thermal inertia
100-$160\,\mathrm{J\,m^{-2}\,K^{-1}\,s^{-1/2}}$, having sublimating
$\mathrm{CO_2}$ ice located $11\pm 4\,\mathrm{cm}$ below the surface. Its
estimated age is consistent with an outburst observed on 2014 April 27-30. The
Hathor site has a $0.8\pm 0.2\,\mathrm{cm}$ dust mantle, a thermal inertia of
$40\pm 20\,\mathrm{J\,m^{-2}\,K^{-1}\,s^{-1/2}}$, no $\mathrm{CO_2}$ ice to
within $1.0\,\mathrm{m}$ depth, and a mantle bulk density of $340\pm
80\,\mathrm{kg\,m^{-3}}$ that is higher than the theoretically expected $180\pm
10\,\mathrm{kg\,m^{-3}}$, suggesting that compression has taken place.
