If spacetime rotates, does that imply there being a centre of the universe?
There's really no way to know if there was something in existence before the big bang however. We just lack evidence of such a thing.
It is true that in GR you can't speak of "before" the big bang, but the big bang itself is a feature within time. It happens at the first moments of time and everywhere in space. And if you replace the initial singularity with a dense quantum foam and thus are able to extend time into the past in some quantum sense, the big bang doesn't go away.
Which would undercut practically all of modern physics. Really fundamental conservation laws like momentum and energy rely on the universe being equal in all directions. If it has a central axis, that does not hold.
So if that holds, it's potentially a major pointer to the very origins of the universe itself. But it's also one of those extraordinary claims that require extraordinary proof. I strongly doubt that this will stand up to scrutiny -- though I'll certainly be pleased if it turns out be true, because that will be a major advance in our understanding.
To say that the universe rotates usually implies that it rotates with respect to something external. If we limit ourselves to the visible universe, this would mean that mass outside our light cones can actually influence us, by means of building the frame of reference that allows us to say that the universe rotates!
It would imply that there exist privileged reference points within the universe, and that would be a major change to physics. I can't predict all of the consequences of that, and they might include some kind of external frame of reference. But that doesn't necessarily follow.
If you have two stars orbiting each other, they orbit a centre of gravity and they will probably both be rotating in the same direction as their orbits.
Is that a meaningful centre for anything else though?