  Floquet higher-order topological insulators and superconductors (HOTI/SCs)
with an order-two space-time symmetry or antisymmetry are classified. This is
achieved by considering unitary loops, whose nontrivial topology leads to the
anomalous Floquet topological phases, subject to a space-time
symmetry/antisymmetry. By mapping these unitary loops to static Hamiltonians
with an order-two crystalline symmetry/antisymmetry, one is able to obtain the
$K$ groups for the unitary loops and thus complete the classification of
Floquet HOTI/SCs. Interestingly, we found that for every order-two nontrivial
space-time symmetry/antisymmetry involving a half-period time translation,
there exists a unique order-two static crystalline symmetry/antisymmetry, such
that the two symmetries/antisymmetries give rise to the same topological
classification. Moreover, by exploiting the frequency-domain formulation of the
Floquet problem, a general recipe that constructs model Hamiltonians for
Floquet HOTI/SCs is provided, which can be used to understand the
classification of Floquet HOTI/SCs from an intuitive and complimentary
perspective.
