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.