  We study translationally-invariant insulators with inversion symmetry that
fall outside the established classification of topological insulators. These
insulators are not required to have gapless boundary modes in the energy
spectrum. However, they do exhibit protected modes in the entanglement spectrum
localized on the cut between two entangled regions. Their entanglement entropy
cannot be made to vanish adiabatically, and hence the insulators can be called
topological. There is a direct connection between the inversion eigenvalues of
the band structure and the mid-gap states in the entanglement spectrum. The
classification of protected entanglement levels is given by an integer $n\in
Z$, which is the difference between the negative inversion eigenvalues at
inversion symmetric points in the Brillouin zone, taken in sets of two. When
the Hamiltonian describes a Chern insulator or a non-trivial T-invariant
topological insulator, the entanglement spectrum exhibits spectral flow. If the
Chern number is zero for the former, or T is broken in the latter, the
entanglement spectrum does \emph{not} have spectral flow, but, depending on the
inversion eigenvalues, can still have protected midgap bands. Although spectral
flow is broken, the mid-gap entanglement bands cannot be adiabatically removed,
and the insulator is `topological.' In 1D, we establish a link between the
product of the inversion eigenvalues of all occupied bands at all inversion
momenta and charge polarization. In 2D, we prove a link between the product of
the inversion eigenvalues and the parity of the Chern number. In 3D, we find a
topological constraint on the product of the inversion eigenvalues indicating
that some 3D materials are topological metals, and we show the link between the
inversion eigenvalues and the 3D Quantum Hall Effect and the magnetoelectric
polarization in the absence of T-symmetry.
