What about angular speed, though? My physics is very rusty where it comes to rotating things, and I don't recall dealing with rotation on relativity classes in school. Since you can derive momentary linear speed from angular speed and distance from centre of rotation, I assume there is no absolute angular speed either?
No, there is absolute angular speed. Which is kind of an odd thing that rotation is like that but linear motion is not. Take the Earth for example - you can't say how fast it's moving unless it's relative to something but you can say it's doing one rotation a day.
Another odd possibly related thing is linear momentum can be any amount but angular momentum is quantized which may be why we have particles. In Maxwell's equations you can have any amount of light but in practice it arrives in chunks with one unit of angular momentum each.
Not really. If you have two bodies in hydrostatic equilibrium arranged so that you can extend the rotational axis of one such that it overlaps completely with the rotational axis of the other, how do you come up with "absolute rotation"?
You could consider the case where one is exactly spherical and the other is highly oblate. While you're free to choose a system of coordinates which keeps the highly oblate one non-rotating with respect to the coordinates, you have to appeal to fictitious forces to explain the oblateness of the coordinate-stationary body and the exact spherical symmetry of the coordinate-rotating body. Physics typically takes a simpler form if you decide the spherical body should be non-rotating against a set of coordinates that covers both bodies. (On the other hand, if these are large bodies -- planets, for example -- and you can put a lab on the surface of each, you get the Special Relativistic form of physics for practically all possible experiments wholly contained within each lab).
However, a more typical case is that neither body is exactly spherically symmetrical, in which case you might want to appeal to the view of the motions of deep sky objects observable on each body. One might be ultra-Machian and say that "absolute rotation is determined by the movement of fixed stars", but really you probably should be more interested about whether you need to resort to pseudotensors in your write-down of local physical behaviours, i.e., rely upon the principle of covariance instead, and ignore arguments motivated by less-formal arguments (sometimes only allegedly) related to any number of things Mach said.
The principle of covariance only allows a picking-out of "absolute rotation" in specific types of universe, and that picking-out is not very useful in a universe like ours.
> angular momentum is quantized
Spin is quantized in the Standard Model. Although one can call spin an intrinsic angular momentum, it's not the same thing as the angular momentum of a rotating macroscopic body like a planet. In particular, intrinsic spin survives changes of coordinates, whereas rotation does not (as discussed above).
There is a (very) technical and quite beautiful overview of the difference here:
