Rotational relativity? Is there an universal frame of reference for rotation? So, there is obviously no such thing as an universal frame of reference for velocity. According to the relativity theory, there is no difference between two observers moving with respect to each other, they both experience identical laws of physics. However, does the same go for rotation?
An object gains rotational energy when torque is applied to it, and it also exerts forces on other objects depending on its rotational velocity. This has been used in many sci-fi movies, where the space-ship has a rotating part (as in the Martian for example), and the astronauts are able to stand casually in there, due to the force that the rotating part exerts on them. Obviously, that would not work if that part of the ship wasn't rotating. However, how do we know whether it is in fact rotating or not? If we are able to tell whether it is, isn't there an universal rotational stationary frame of reference, to which we have to compare everything that rotates?
 A: A rotating reference frame is not an inertial reference frame: In the rotating frame, objects accelerate even though there are no forces acting on them.
In your example, you can in fact determine easily whether you are rotating or the universe is rotating around you. In the first case there is artificial gravity on the ship, and in the second case there is not.
So yes, rotation works differently from velocity. There is not one "universal stationary frame of reference" though, there are many: The class of reference frames that are neither rotated nor accelerated in any other way are the inertial frames.
A: The other answers to this question do all follow the classic line of thought: we can differentiate the two following situations:


*

*(i) we rotate with respect to a non-rotating universe,

*(ii) we don't rotate but the universe rotates around us.


The idea being that in case (i) we would observe Coriolis and centrifugal force (the latter being the one the OP wrote about when mentioning scifi contraptions) but not in case (ii).
That debate goes all the way back to at least Newton, and it was key in the thinking of Mach that inertia should be entirely explainable by gravity. A very good historical account can be found in [Pfi07], which is very readable by non-experts (among which I place myself!). This articles shows how physicists have slowly come to the realisation through the 20th century that a rotating nearly spherical shell of matter induces the Coriolis and the centrifugal force inside itself, and exactly so (the mathematical demonstration of the result can be found in a classic paper by the same author and a collaborator [PB85]). By induce, I mean the gravitational effect of the rotating shell: I am not assuming that everything inside the shell rotates along with it (that would be circular a reasoning!). I also mean that those inertial forces are generated everywhere inside that shell, not just near the centre (which is an earlier less powerful result you are likely to stumble upon).
Even though this is not realistic at all, it should give us pause, as it completely contradicts the classic line of thought I reminded above: case (ii) does exactly reproduce case (i)!! Now, one may ask with respect to what this shell of matter rotates. The answer is that one postulates a spacetime asymptotically flat: in simple words, infinitely far, Newton first principle holds good, i.e. a test mass is not subject to any gravitational or inertial forces. However inside the shell, an observer observing the trajectories of moving bodies would come to the conclusion that he is tied to a rotating frame even though he is not.
I do not claim this is in any way the "ultimate" answer but this is very interesting food for thought…
[Pfi07] Herbert Pfister. On the history of the so-called lense-thirring effect. General Relativity and Gravita- tion, 39(11):1735–1748, Nov 2007. Free access on citeseerx
[PB85] H Pfister and K H Braun. Induction of correct centrifugal force in a rotating mass shell. Classical and Quantum Gravity, 2(6):909, 1985.
A: I'm going to say no--there is not a universal "at rest" frame for rotation. Certainly here in Earth we can measure our rotation, and find a frame fixed relative to the universe. Moreover, other people around the Solar System might agree--unless they notice the minute frame-dragging (aka Lens-Thirring effect).
Near a rotating black hole the effect can be large, and if you're in you space ship feeling no centripetal force [that is, you're officially not rotating]--observers back on Earth would see you rotating, and vice versa.
Also: the frame dragging can be differential--so 2 nearby points define "not rotating" differently. This gives rise to so-called vortex-lines for visualizing space-time near black holes:
http://www.caltech.edu/news/physicists-discover-new-way-visualize-warped-space-and-time-1680
A: Yes, there is a universal reference frame for rotation. One could also make the trivial argument that there are infinitely many such reference frames, all of which would have zero angular velocity with respect to each-other and the overall long range structure of the universe.
A rotating object (or a set of rotating objects) that is(are) somehow held in rotation around a point by a force other than solely by the mutual gravitational attraction is(are) not moving freely along a geodesic [ https://en.wikipedia.org/wiki/Geodesics_in_general_relativity ] - there is constant acceleration resulting from a force that keeps the object(s) along the rotational paths, commonly referred to as the centripetal force. This could be in the form of the tension in a string or other mechanical linkage keeping multiple objects together as they rotate around each other, or in the case of a single object that deforms (bulges) - e.g., a sphere that bulges at the equator - it is the cohesiveness of its material providing the necessary stabilizing force to ensure that it doesn't fall apart.
Within the reference frame of the objects that are rotating around each-other (or pieces of the object that is rotating around its axis of rotation), a corresponding centrifugal force is felt, which would otherwise not have been felt if there was no such rotation.
The presence of this centrifugal force indicates that the object(s) is(are) rotating. The rotation can be considered to be with respect to the overall long range structure of the universe. There is some debate about how to define that exactly [ https://en.wikipedia.org/wiki/Absolute_rotation ] -  but in general, the presence of a centrifugal force that appears to pull rotating object(s) apart is considered to be a sign that the object(s) in question is(are) rotating with respect to an absolute frame of reference. 
In the special case of two or more gravitationally bound objects rotating about their common center of mass, these objects (any any observers on them) are actually moving freely along a geodesic and the gravitational force acting on an observer located on one of these objects (e.g., an astronaut on the ISS orbiting the earth) would exactly balance out any centrifugal force that would be expected to arise within the reference frame of such an observer.
The explanation provided in arXiv:physics/0409010 is fairly accessible to the general audience.
A: Under GR gravitational and accelerated relativity rotational motion is not an inertial frame of reference. An inertial frame of reference in GR must have a constant velocity both speed and direction of motion. For GR acceleration means that is, it is moving at an accelerated speed in a straight line, or at fixed speed with varying direction or both.
However what SR concerns, rotational motion at constant speed is an inertial frame of reference. The constant change in direction has no effect on SR time dilation and only the constant speed value counts for the phenomenon. Also a fixed speed circular rotation does not generate any GR acceleration time dilation as demonstrated in the recent g-2 Fermilab muon experiment and only SR constant circular speed time dilation was measured. Which proves my point.
A: Attracting GRT
Light trajectories describe the geodesics of space according to the field equation. Assuming independence of rotational frames and the universe rotating relative to geodesics of space and a fix light source sending out beams, the universe would observe the trajectories as spirals. We don't observe this!
A: Thought Experiment inside SRT:

*

*Let the speed of light be invariant

*Assume equality of rotational systems

*Assume, rotation does not change the time base

Consider an optical system, that lets a beam rotate around an object on a circle (or regular polygon with sufficient legs) with fix radius r. Now, we expect that all assumptions hold in both systems, independent from rotation.
Let the universe including the optical system rotate around the fix object with angular speed $\omega$. As the optical system does not move in direction of the radius, no length contraction applies to the radius. Due to the invariant time base [3], the period T of the beam remains the same.
In the case, that universe and beam rotate in the same direction, the path length of the beam after period T is [1]
$c\cdot T = (2\pi-\omega T) r$
As $c,T,r$ are invariant, any $\omega\neq 0$ leads to a contradiction.
Conclusion: SRT does not support equality of rotational systems.
