  The complete characterization of a generic $d$-dimensional Floquet
topological phase is usually hard for the requirement of information about the
micromotion throughout the entire driving period. In a recent work [L. Zhang et
al., Phys. Rev. Lett. 125, 183001 (2020)], an experimentally feasible dynamical
detection scheme was proposed to characterize the integer Floquet topological
phases using quantum quenches. However, this theory is still far away from
completion, especially for free-fermion Floquet topological phases, where the
states can also be characterized by $Z_{2}$ invariants. Here we develop the
first full and unified dynamical characterization theory for the $Z_{2}$
Floquet topological phases of different dimensionality and tenfold-way symmetry
classes by quenching the system from a trivial and static initial state to the
Floquet topological regime through suddenly changing the parameters and turning
on the periodic driving. By measuring the minimal information of Floquet bands
via the stroboscopic time-averaged spin polarizations, we show that the
topological spin texture patterns emerging on certain discrete momenta of
Brillouin zone called the $0$ or $\pi$ gap highest-order band-inversion
surfaces provide a measurable dynamical $Z_{2}$ Floquet invariant, which
uniquely determines the Floquet boundary modes in the corresponding quasienergy
gap and characterizes the $Z_{2}$ Floquet topology. The applications of our
theory are illustrated via one- and two-dimensional models that are accessible
in current quantum simulation experiments. Our work provides a highly feasible
way to detect the $Z_{2}$ Floquet topology and completes the dynamical
characterization for the full tenfold classes of Floquet topological phases,
which shall advance the research in theory and experiments.
