  We study the multi-gap topology of the periodic spectra of Wilson loop
operators (WLOs) in mirror symmetric insulators. We develop two topological
invariants each associated with a mirror-invariant gap in the Wilson loop
spectrum. We propose that both topological invariants in combination determine
the general higher-order bulk-boundary correspondence in 2D mirror symmetric,
boundary-obstructed topological insulators. Finally, we demonstrate that these
new multi-gap topological invariants apply to anomalous cases beyond those
captured by the nested Wilson loop, and we subsequently develop an
understanding of the correlation between WLOs along two orthogonal directions.
