I think that, just like all motion, rotation is relative, but I may be wrong.
Every point on a rotating body thinks the body is rotating around IT, but it's only the point at the center of mass (or the pivot) which feels like its stationary relative to the universe. Let me give an example:
Imagine this circle is the entire universe. There's nothing more than this circle:
Okay, now there's something more than the circle. The god of physics stack exchange said let there be a rod, and now you also see a single black horizontal rod.
Each of the colored blobs on the rod is a different point which the rod could rotate around. This view is from the point of view of a hyperdimensional being, watching from the outside of the universe (outside the universe because they're staring down at the paper).
The rod could rotate around the red point....
Around the pink point...
or around the blue point...
Now, let's consider ourselves as being really small. Extremely small. We shrink smaller and smaller, and all of a sudden are teleported to the pink point.
Recall that the circle represents the entire universe the rod exists in; there is nothing more but the rod and the circle (ring, universe, whatever). Period.
Oh, and there's also no gravity. And no oxygen. We don't need to breathe.
Now, the rod rotates “around” the blue point. From the point of view of an otherworldly outsider that exists outside the universe, that's exactly what it does. It rotates around the blue point.
But what about us, standing on the pink point? If the rod rotates fairly quickly, we'll feel an increase in the normal force from the rod. And thus we'll be able to tell its the rod that's moving and not the world around us. (Ignore the picture in bottom right, that was a mess-up)
BUT, IF the rod rotates really slowly... then what we might really think is that the rod rotated around us (around the pink point) counter-clockwise, but simply also pulled the ring (universe) along with it! There really is no difference!
We would actually think the rod rotated around us, but it was simply attached to the circle at the blue point, attached to the universe at the blue point, and looks like this:
We can do this for the other scenarios as well. In fact, for every rotation, we can pick any arbitrary point on the rod and say that the rod rotated arount it, its just only ONE of those points will think the universe was still as well. All the others will simply think the universe was attached to the rod at some other point, and rotated with it.
Let's look at a rotation where we agree with the hyperdimensional viewer staring down at the paper (if the circle is the entire universe...what's the paper?!).
In the rotation where (to the viewer looking at the paper) the rod rotates “around” the pink point, to us (“standing at the pink point”) the rod rotated around the pink point as well!
In this rotation, the universe was still relative to us (to the pink point). And we thought the rod rotated around us.
But these are two separate statements. We could've thought the rod rotated around us in any scenario, simply only in one will the universe be still relative to us as well.
This was the difference between this and the last rotation: that in this rotation, the universe was still relative to us, while in the last one, we saw it move with the rod. But we think that the rod rotated around us regardless of the situation, simply that in the first scenario the rod was for some reason attached to the circle (universe) at the blue point and thus it rotated around us as well.
But what if there was no universe? What if there was no circle?
If there was no universe for the rod to rotate in, from the point of view of the person looking down we would be the only point of reference, they may very well agree that the rod rotated around us (pink point) in the first scenario as well! Then if we add the ring, they may agree that the ring was connected to the blue point, and got pulled along with it!
Rotation is relative
Finally, when the rod rotates “around” the red point from the point of view of the hyperdimensional viewer, to us (“standing at the pink point”) everything rotated around us again, we were the center of rotation, and it's just the red point pulled the universe with it upwards and to the left!
Either way, in all three scenarios, we can say the rod spun around an arbitrary point on the rod. That is, it spun around us. It's just in some scenarios, the universe followed along.
We are the center of everything, and the universe spins around us.
But if the universe wasn't there to begin with, then yes, there would be no circle to rotate relative to, and there would indeed be no way to tell which point the rod was rotating about.
Happy thoughts! The universe rotates around you!
Hope that helped.