  We classify chiral symmetric periodically driven quantum systems on a
one-dimensional lattice. The driving process is local, can be continuous or
discrete in time, and we assume a gap condition for the corresponding Floquet
operator. The analysis is in terms of the unitary operator at a half-period,
the half-step operator. We give a complete classification of the connected
classes of half-step operators in terms of five integer indices. On the basis
of these indices it can be decided whether the half-step operator can be
obtained from a continuous Hamiltonian driving, or not. The half-step operator
determines two Floquet operators, obtained by starting the driving at zero or
at half period, respectively. These are called timeframes and are chiral
symmetric quantum walks. Conversely, we show under which conditions two chiral
symmetric walks determine a common half-step operator. Moreover, we clarify the
connection between the classification of half-step operators and the
corresponding quantum walks. Within this theory we prove bulk-edge
correspondence and show that a second timeframe allows to distinguish between
symmetry protected edge states at $+1$ and $-1$ which is not possible for a
single timeframe.
