Relativity doesn't apply to acceleration. If there were two bodies in the universe, and one accelerated away from the other, it doesn't depend on perspective. Each body is capable of independently determining whether it was the one accelerating, and both would come to the same conclusion.

The same applies to rotation. Accelerometers placed on my floor and on my body will detect the difference, and both will agree.

Special relativity says that there is no privileged perspective with respect to linear velocity, but general relativity showed that rotating reference frames are different, since one frame sees a centripetal forces while the other frames sees a centrifugal/gravitational force.

To put it differently, if I pass someone on the train moving at constant velocity, there's no experiment that I can perform which proves that I'm moving while the other person is standing still, or vice versa.

However, if I'm in a centrifuge, instead of on a train, then I can figure out which one of us is moving by measuring who is pressed up against the wall and making funny faces.

Related - Mach's principle:

http://en.wikipedia.org/wiki/Mach's_principle

An important implication of relativity is that when you push something, you do not change it's velocity. You change it's acceleration. It's a small but important distinction. Acceleration is more fundamental.

My interpretation of that quote is that he is referring to his impression after he stops. That's why he specifically references the stopping. You may be familiar with the experience - twirl around for a little bit then stop, and you will feel like the world ("entire universe"?) is still spinning. Even though it's not.

Of course, I may be ignorant of his actual intent.

No, I think the point is that your mind interprets the universe as spinning when you stop twirling. This is just an interpretation, as is the flow of time.

Semantics makes that idea meaningless. "Interpreting" implies time flowing, so its circular.

It doesn't need the "and stop" - a rotating reference frame is not inertial.