Yes. And the requirement for this is that there is a force acting upon the object perpendicular the object's velocity.The simplest case would be an object that takes a circular path.
For a universal spin, all mass in the universe must also experience this force, perpendicular to it's velocity and inversely proportional to the square of the distance from the center of rotation (barycenter, if you will, similar to solar systems, galaxies, etc.).
This implies that there is a point in the universe that is different for all the other points in the universe. This would be that barycenter, and a mass located at this point would not feel the force felt by all other objects in the universe.
From this point, the universe would look the same in all directions -- "smooth", with no significant variations in mass, or energy, or temperature.
Just like it appears from here on Earth.
It has been argued (successfully IMHO) that every location in the universe can -- likewise -- look like that barycenter. This is a contradiction that I believe proves a barycenter does not exists, and therefore there is no universal rotation.
I believe this is a consequence of the fact that our universe is larger than the distance light has traveled since the start of the universe, and therefore, "no matter where you go, there you are" -- all points in the universe look self-similar.
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Alternatively, one could suggest that all matter in the universe has an "average rotation" or spin, and that the cumulative spin of all the matter in the universe sum to a particular non-zero spin value.
I cannot argue that there is some total non-zero spin of the universe. What I know is that such a spin is unmeasurable -- it is impossible to include matter that is outside our light cone (a majority of the mass in our universe). If it cannot be measured, it cannot have an effect, and therefore does not, for all intents and purposes, exist.
There is ***much*** more I could say about this, but I'm outta time -- wheels gotta keep spinning.


