  We benchmark angular-momentum projected Hartree-Fock calculations as an
approximation to full configuration-interaction results in a shell model basis.
For such a simple approximation we find reasonably good agreement between
excitation spectra, including for many odd-$A$ and odd-odd nuclides. We
frequently find shape coexistence, in the form of multiple Hartree-Fock minima,
which demonstrably improves the spectrum in the $sd$- and $pf$-shells. The
complex spectra of germanium isotopes present a challenge: for even $A$ the
spectra are only moderately good and those of odd $A$ bear little resemblance
to the configuration-interaction results. Despite this failure we are able to
broadly reproduce the odd-even staggering of ground state binding energies,
save for germanium isotopes with $N > 40$. To illustrate potential
applications, we compute the spectrum of the recently measured dripline nuclide
$^{40}$Mg. All in all, projected Hartree-Fock often provides a better
description of low-lying nuclear spectra than one might expect. Key to this is
the use of gradient descent and unrestricted shapes.
