Les,
After some re-reading and thinking about the article, I am going to argue against the article's conclusion. I think the experiment can be paraphased as follows.
The eyes are consciously forced to fix on a stationary and unseen target, while the secured head, neck and body, are put through a sequence of rotations whose angles do not commute.
The argument is the brain must keep the eyes locked on target through a series a motor commands which can not be commutative, because the relative rotations of their reference frame (head, neck, and body) do not commute.
The experiment utilizes the equivalence principle between inertial frames, in which the eyes are fixed relative to the body, while the body rotates through various angles around them. Equivalently, the eyes move around through various angles inside a fixed head.
The experiment is said to demonstrate that the Vestibular-ocular reflex requires non-commutative neural activity. In a technical sense the article only demonstrates that someone's commutative model of this reflex doesn't work, and the authors non-commutative model does work. But let's skip that detail.
In order to exchange valid rules of motion between inertial frames, the transforms must be a set of invariants. In this case the neural circuitry performs an equivalent function to that of the invariant transform between inertial frames. In the above experimental context, the question is posed, are these invariants themselves commutative or non-commutative?
The invariants have to be commutative in order to faithfully preserve under transformation either the commutative or non-commutative relations of the rules exchanged.
So, the demonstrated fact that a commutative model fails, implies there is something wrong with the model. But, whether or not its commutativity is involved is indeterminate.
Further, if my characterization of the experiment as an alias/alibi problem between coordinates and origin (move origin <=> move coordinates) is correct, then the transforms must be commutative.
This argument only says that our kinesthic perception faithfully transmits the commutative and non-commutative features of the space around us and our motions through it.
Chuck Grimes