[soVeryTired]

Rotational invariance makes sense if you're free to pick the scale and direction of your coordinates. If that's the case then a large class of distributions can be made rotationally invariant (at a guess, I'd say all elliptical distributions [0], but I don't have a handy proof)

Independence of (x, y) components is a trickier one to justify IMO. Something like a Cauchy or Student t model could be made rotationally invariant, but the coordinates are no longer independent.

A Student t distribution behaves like a standard Gaussian, but with a second source of randomness that controls the size of its radial distance from the origin. So you get darts that are often on-target, but with occasional huge misses. If the X component is gigantic, that suggests that the Y component is also large, breaking independence.

[0] https://en.wikipedia.org/wiki/Elliptical_distribution