  Periodically driven systems with internal and spatial symmetries can exhibit
a variety of anomalous boundary behaviors at both the zero and $\pi$
quasienergies despite the trivial bulk Floquet bands. These phenomena are
called anomalous Floquet topology (AFT) as they are unconnected from their
static counterpart, arising from the winding of the time evolution unitary
rather than the bulk Floquet bands at the end of the driving period. In this
paper, we systematically derive the first and inversion-symmetric second-order
AFT bulk-boundary correspondence for Altland-Zirnbauer (AZ) classes BDI, D,
DIII, AII. For each AZ class, we start a dimensional hierarchy with a parent
dimension having $\mathbb{Z}$ classification, then use it as an interpolating
map to classify the lower-dimensional descendants. From the Atiyah-Hirzebruch
spectral sequence (AHSS), we identify the subspace that contains topological
information and faithfully derive the AFT bulk-boundary correspondence for both
the parent and descendants. Our theory provides analytic tools for
out-of-equilibrium topological phenomena.
