  In this article we give a comprehensive treatment of a `Clifford module flow'
along paths in the skew-adjoint Fredholm operators on a real Hilbert space that
takes values in KO${}_{*}(\mathbb{R})$ via the Clifford index of
Atiyah-Bott-Shapiro. We develop its properties for both bounded and unbounded
skew-adjoint operators including an axiomatic characterization. Our
constructions and approach are motivated by the principle that \[
  \text{spectral flow} = \text{Fredholm index}. \] That is, we show how the
KO--valued spectral flow relates to a KO-valued index by proving a
Robbin-Salamon type result. The Kasparov product is also used to establish a
spectral flow $=$ Fredholm index result at the level of bivariant K-theory. We
explain how our results incorporate previous applications of $\mathbb{Z}/
2\mathbb{Z}$-valued spectral flow in the study of topological phases of matter.
