Question: Is the Universe rotating around its center?
Discussion: Relatively recently astronomical physicists discovered that the rate of expansion of the universe is increasing. Astronomical physicists also discovered that the universe may be relatively flat and commonly referred to as the "axis of evil". [1] Both of these discoveries may be consistent with a universe that rotates around a common axis.
"Dark energy" has been proposed to account for the increasing rate of expansion of the universe. However, "Dark Energy" may be no more mysterious than common centrifugal forces caused by the rotation of the universe around its common center of gravitational mass, but the math has not been done to confirm or deny this conjecture.
Conclusion: A rotating universe may be consistent with the discovery that the universe may be flat and the expansion rate of the universe is increasing. "Dark energy" may be no more mysterious than common centrifugal forces, but the math has not been done to confirm or deny this conjecture.
[1] NewScientist (Zeeya Merali), 4/13/2007, 'Axis of evil' a cause for cosmic concern, NewScientist Magazine issue 2599.