  The ground state of translationally-invariant insulators comprise bands which
can assume topologically distinct structures. There are few known examples
where this distinction is enforced by a point-group symmetry alone. In this
paper we show that 1D and 2D insulators with the simplest point-group symmetry
- inversion - have a $Z^{\geq}$ classification. In 2D, we identify a relative
winding number that is solely protected by inversion symmetry. By analysis of
Berry phases, we show that this invariant has similarities with the first Chern
class (of time-reversal breaking insulators), but is more closely analogous to
the $Z_2$ invariant (of time-reversal invariant insulators). Implications of
our work are discussed in holonomy, the geometric-phase theory of polarization,
the theory of maximally-localized Wannier functions, and in the entanglement
spectrum.
