In Newtonian physics, rotary motion is absolute even though linear motion is not. More recently, there has been a hope that rotation can also be shown to be relative.
Two definitions of relative
A hang up in discussing this question is what I see as two different definitions of "relative."
Definition 1: A quantity is relative if you have to measure it relative to something. I think this definition is a back-definition (backronym) and a play on English words. It doesn't capture the concept of relativity because one can say, "Aristotle thought motion is relative since it must be measured relative to the firmament." That is absolutism, not relativity.
Definition 2: A quantity is relative if you can imagine it being vastly different and expect to observe the same behavior.
Velocity is relative according to the 2nd definition because we can imagine ourselves and the Earth traveling at relativistic speed and we will make the same predictions. We can imagine ourselves distorted by special relativity, and this won't change our behavior.
In General Relativity, other theories of gravity, modern theories like String Theory, and any others worth mentioning, to what extend is rotation relative according to the 2nd definition?
For the 2nd definition to have meaning, there must be a concept of space separate from the matter of the universe--a way to imagine rotation where there is none.
If rotation were relative, we could imagine the whole universe spinning at the same rate as an object and not observe any new forces acting on the object. If rotation were relative, we could imagine that a spinning object were in fact stationary and the whole universe were spinning around it, with the same centrifugal forces acting on the object.