I believe they are all rotated by the same random matrix, the purpose being (IIUC) to distribute the signal evenly across all dimensions. So effectively it drowns any structure that might be present in noise. That's essential for data efficiency in addition to avoiding bias related issues during the initial quantization step. However there are still some other issues due to bias that are addressed by a second quantization step involving the residual.

That said, I don't believe the visualization is correct. The grid for one doesn't seem to match what's described in the paper.

Also it's entirely possible I've misunderstood or neglected to notice key details.