  Building from work by Cedzich et al. and Suzuki et al., we consider
topological and index-theoretic properties of chiral unitaries, which are an
abstraction of the time evolution of a chiral-symmetric self-adjoint operator.
Split-step quantum walks provide a rich class of examples. We use the index of
a pair of projections and the Cayley transform to define topological indices
for chiral unitaries on both Hilbert spaces and Hilbert $C^*$-modules. In the
case of the discrete time evolution of a Hamiltonian-like operator, we relate
the index for chiral unitaries to the index of the Hamiltonian. We also prove a
double-sided winding number formula for anisotropic split-step quantum walks on
Hilbert $C^*$-modules, extending a result by Matsuzawa.
