Periodicity modulo 8 appears in the classification of real Clifford algebras $C\ell_{p,q}(\mathbb{R})$ (usualy refered to as the "Clifford Clock"), in real Bott periodicity and in the definition of a real structure of KO-dimension on a spectral triple. The latter concept can be found in Connes-Marcolli book http://alainconnes.org/docs/bookwebfinal.pdf, for instance.
Spectral triples are a generalization of spin$^c$ manifolds and real spectral triples of spin manifolds. In fact, every (real) spectral triple over a commutative $*$-algebra is a spin manifold, by certain reconstruction theorems proven by Connes and, independently and under other conditions, by A.Rennie and J.Várilly. The KO-dimension $N\in\mathbb{Z_8}$ of a real spectral triple is enterly determined by knowing whether certain operators on a Hilbert space $H$ commute or anticommute. $H$ generalizes the square-integrable spinors Hilbert space.
Being alien to K-theory, I suspect that the definition of KO-dim is motivated (as many concepts in noncommutative geometry are) by what happens in the "commutative case" (spin geometry). I want to know where do such commutation and anticommutation relations appear in KO-theory. Otherwise put, what is the motivation for the definition of KO-dim, from the point of view of K-theory? can this periodicity be related to real Bott periodicity or the periodicity of the Clifford clock?