  Floquet engineering provides a toolbox for the realization of novel quantum
phases without static counterparts, while conventionally the realization may
rely on the manipulation of complex temporal evolution. Here we propose a
systematic and high-precision scheme to realize unconventional Floquet
topological phases by engineering local band structures in particular momentum
subspace called band inversion surfaces (BISs). This scheme is based on a new
bulk-boundary correspondence that for a class of generic $d$-dimensional
periodically driven systems, the local topological structure formed in each BIS
uniquely determines the features of gapless boundary modes. By engineering the
BIS configuration we demonstrate a highly efficient approach to realize,
manipulate, and detect novel Floquet topological phases. In particular, we
predict a two-dimensional (2D) anomalous Floquet valley-Hall phase which
carries trivial global bulk topological invariants but features protected
counter-propagating edge states in each quasienergy gap. The unconventional
nature of this novel 2D phase is further illustrated by the examination of edge
geometry dependence and its robustness to disorder scattering. Anomalous chiral
topological phases with valley protection in higher dimension are also
predicted and studied. Our systematic and highly feasible scheme opens a new
route to realize and engineer unconventional Floquet topological phases for
ultracold atoms and other quantum simulators.
