  Inspired by a recently constructed commuting-projector Hamiltonian for a
two-dimensional (2D) time-reversal-invariant topological superconductor [Wang
et al., Phys. Rev. B 98, 094502 (2018)], we introduce a commuting-projector
model that describes an interacting yet exactly solvable 2D topological
insulator. We explicitly show that both the gapped and gapless boundaries of
our model are consistent with those of band-theoretic, weakly interacting
topological insulators. Interestingly, on certain lattices our
time-reversal-symmetric models also enjoy $\mathcal{CP}$ symmetry, leading to
intuitive interpretations of the bulk invariant for a $\mathcal{CP}$-symmetric
topological insulator upon putting the system on a Klein bottle. We also
briefly discuss how these many-body invariants may be able to characterize
models with only time-reversal symmetry.
