  Realising photonic analogues of the robust, unidirectional edge states of
electronic topological insulators would improve our control of light on the
nanoscale and revolutionise the performance of photonic devices. Here we show
that new symmetry protected topological phases can be detected by reformulating
energy eigenproblems as Berry curvature eigenproblems. The "Berry bands" span
the same eigenspace as the original valence energy bands, but separate into
pseudo-spinful and pseudo-spinless subspaces in
$\mathrm{C}_2\mathcal{T}$-symmetric crystals. We demonstrate the method on the
well-known case of Wu & Hu [Phys. Rev. Lett. 114, 223901 (2015)] and a recently
discovered fragilely topological crystal, and show that both crystals belong to
the same $\mathrm{C}_2\mathcal{T}$-protected $\mathbb{Z}_2$ topological phase.
This work helps unite theory and numerics, and is useful in defining and
identifying new symmetry-protected phases in photonics and electronics.
