That actually depends on the action. The one we're using mostly in our example, the forward half of a natural, is a good example of this. However, there should be some curve of direction on the CBM on the back half of a natural, as well as either half of the reverse (the back half having the stronger curve of direction). If both parties move on a straight line, they rotate independently, rather than rotating around a common center, and the dance position is altered.

At any rate, the curving of direction is probably the one aspect of CBM that doesn't actually swing, because the direction that the center travels when curving is hardly a release of energy through that particular arc. So that point may be irrelevant to this discussion, but I do mention it because we don't want to mislead people into thinking that all (or even most) CBM movements occur on a straight line. But let's assume that they do anyway, or at least look specifically at the one example where the direction is straightest, the forward half of the natural.

You make the point that the arc of swing is not literally about a fixed axis outside of the body (as though you were hanging onto a pole and circling around it), but that your body just happens to be rotating while traveling a straight line, and I do concur (at least with the forward half of natural). But that doesn't mean it can't still be thought of as a rotary swing. You can stand in place and rotate, and if you do so with enough energy to release (in other words, enough that momentum takes over what the muscles began), you're producing rotary swing. It's a relatively small arc, but then swing is not defined by the size of the arc.

The fact that you are also traveling forward broadens the arc through space, but does not affect the axis. You are not swinging around a remote axis, but rather swinging around the same axis you would be even if you weren't traveling... the difference is, now it's a traveling axis.

None of this, however, prevents people from feeling as though they are swinging around a remote axis (fixed point outside of the body). There's nothing wrong with having that impression, even if it's not an accurate description of the true physics behind it.

Take, for example, the idea of pendular swing. The very idea is ridiculous. You can't have true pendular swing unless you hang from a fixed point above. I don't know anybody who swings from a vine while ballroom dancing, but that still doesn't stop people from describing the action as "pendular". And that's perfectly fine, if it produces the right results.

Here's my point: I can't disagree with physics. But whether the axis of rotation is real or imaginary is not as important as whether the impression it leaves on a dancer is the right one to produce the intended result.

Regards,
Jonathan