  We consider the extent to which symmetry eigenvalues reveal the topological
character of bands. Specifically, we compare distinct atomic limit phases (band
representations) that share the same irreducible representations (irreps) at
all points in the Brillouin zone and, therefore, appear equivalent in a
classification based on eigenvalues. We derive examples where such
"irrep-equivalent" phases can be distinguished by a quantized Berry phase or
generalization thereof. These examples constitute a generalization of the
Su-Schrieffer-Heeger chain: neither phase is topological, in the sense that
localized Wannier functions exist, yet there is a topological obstruction
between them. We refer to two phases as "Berry obstructed atomic limits" if
they have the same irreps, but differ by Berry phases. This is a distinct
notion from eigenvalue obstructed atomic limits, which differ in their symmetry
irreps at some point in the Brillouin zone. We compute exhaustive lists of
elementary band representations that are irrep-equivalent, in all space groups,
with and without time-reversal symmetry and spin-orbit coupling, and use group
theory to derive a set of necessary conditions for irrep-equivalence. Finally,
we conjecture, and in some cases prove, that irrep-equivalent elementary band
representations that are not equivalent can be distinguished by a topological
invariant.
