  Motivated by the fact that a polarized ${}^3$He nucleus behaves as an
`effective' neutron target, we examine manifestations of neutron
electromagnetic polarizabilities in elastic Compton scattering from the
Helium-3 nucleus. We calculate both unpolarized and double-polarization
observables using chiral perturbation theory to next-to-leading order
(${\mathcal O}(e^2 Q)$) at energies, $\omega \leq m_{\pi}$, where $m_{\pi}$ is
the pion mass. Our results show that the unpolarized differential cross section
can be used to measure neutron electric and magnetic polarizabilities, while
two double-polarization observables are sensitive to different linear
combinations of the four neutron spin polarizabilities.
  [Note added in 2018] The qualitative conclusions and analytic formulae
presented in this paper are correct, but several of the numerical results are
wrong: see the erratum posted as arXiv:1804.01206 for further details. A full
suite of corrected numerical results for cross sections and asymmetries can be
found in Margaryan et al., arXiv:1804.00956. They can also be obtained as an
interactive Mathematica notebook by emailing<EMAIL_ADDRESS>