  Discrete-time quantum walks (DTQW) have topological phases that are richer
than those of time-independent lattice Hamiltonians. Even the basic symmetries,
on which the standard classification of topological insulators hinges, have not
yet been properly defined for quantum walks. We introduce the key tool of
timeframes, i.e., we describe a DTQW by the ensemble of time-shifted unitary
timestep operators belonging to the walk. This gives us a way to consistently
define chiral symmetry (CS) for DTQW's. We show that CS can be ensured by using
an "inversion symmetric" pulse sequence. For one-dimensional DTQW's with CS, we
identify the bulk ZxZ topological invariant that controls the number of
topologically protected 0 and pi energy edge states at the interfaces between
different domains, and give simple formulas for these invariants. We illustrate
this bulk--boundary correspondence for DTQW's on the example of the "4-step
quantum walk", where tuning CS and particle-hole symmetry realizes edge states
in various symmetry classes.
