If you find that the galaxies rotate around you, then the universe is said to be rotating. You have actually only seen that the universe rotates around the point where you are, but if the Copernican principle holds, then it rotates around any point. That's a rotating universe. So keep in mind that when I talk about a rotating universe, I mean that the matter of the universe rotates around the non-rotating observer. The Earth is rotating.
Centrifugal forces flatten the Earth a little.
Also the Coriolis forces, having a strong impact on winds and missiles, indicate that the Earth is rotating. It is a postulate that there exists such a system. At least rough measuremens seem to indicate that the Machians are right in their prediction, and thus that the inertial system of the Newtonians in some magic way appears to be aligned with the masses of the Universe. Strange coincidence, isn't it, for the Newtonians, as in Newtonian theory there is no connection with the reference systems and the masses that move around in them. If the Machians are right, then of course rotation of the universe is impossible.