Most objects in the universe cycle around their centre of mass: earth, sun, galaxies. Even galaxy clusters.
However, the whole universe doesn't possess an angular momentum. It can't: If it had one, the centre would be a special point (which isn't allowed) and the velocities of the outer regions would easily become infinite.
How does the angular momentum vanish with increasing scale?
Shouldn't the laws of nature that lead to cycling for smaller objects also hold for larger objects?
… the whole universe doesn't possess an angular momentum. It can't: If it had one, the centre would be a special point (which isn't allowed) and the velocities of the outer regions would easily become infinite.
This is wrong. It is possible for the universe to rotate around every point of space, without having a special center of rotation, similar to how empirical Hubble's law does not mean that our position is the special center of expansion.
There are solutions of general relativity such as the Gödel metric that describe precisely this: a homogeneous universe uniformly rotating around its every point.
The term “angular momentum of the universe” may not have a well–defined meaning (see this question for a related discussion of the “energy of the universe”), but the “angular velocity of universe's rotation” is quite an intuitive characteristic on par with Hubble's parameter.
For a relatively recent review on the subject of universe's rotation see this paper.
