The cross product of Quaternions can be right-handed or left-handed; that of Octonions more complicated ("untwisted" or "twisted"). The octonions are comprised of 7 overlapping quaternionic triads, each of which can have a right- or left-handed crossproduct. If certain interrelationships among the handednesses of these triads are not observed, one obtains the Twisted Octonions, rather than the usual Octonions. The Twisted Octonions have a natural decomposition into a 4-vector and a spinor, suggesting a relation to superspace and supersymmetry.
Perhaps there is also a relationship to the body-mind and matter-spirit dualities. We understand the Universe as an assemblage of material objects in 4-dimensional Space-Time. If Space-Time- is actually 8-dimensional, with extra "spin" coordinates, then all phenomena have extra degrees of freedom, and the biochemical phenomena of the brain, as well as the self-organizing principle powering biological evolution, might have geometrical structure in the extra dimensions that causes Consciousness to permeate all things.