Wouldn't the center be the Big Bang and the 3D Universe at the current time (if Relativity lets me write about a current time) be the 3D surface of the 4D sphere (or spheroid) that the Big Bang is creating by keeping to expand?

the big bang happened everywhere

With the caveat that by some awkward metric 'everywhere' was much smaller when it happened.

Well, that depends whether the universe is infinite. In any case, everything was closer together and hotter.

Infinities come with a spectrum, some are smaller than others . . .

and some universes are finite.

It's more correct to just say location had no meaning until something existed in 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.

I think "the big bang happened everywhere" is more correct than your take. The big bang dlrefers to the early period of a spacetime with an initial moment. At that initial moment the distance between all points is zero. But there still are a full 3d set of locations. Starting at this initial moment, distances between points grow quickly. Thats the bang.

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.

Yes, it would. Or at least a central axis.

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.

If everything started from a singularity that had a axiomatically uniform rotation, you might not be able to assess the center "axis" based from inside spacetime itself

This seems to be a variation of Mach's principle, which is one of the closest things to philosophy in physics!

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!

You don't need any external reference points to know that you're rotating. You could detect it by the apparent centrifugal force: objects in your own frame of reference don't move in straight lines.

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.

There is. The center of the observable universe is earth. Every alien civilization will see themselves at the center of thier observable universe. And each will observe the same rotation in faraway objects. Relativity makes things strange.

Depends what you mean by centre, I guess.

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?

If the universe is the interior of a black hole, one would assume so.

And an axis.

Or an axis.

And my ax