That's not quite what "rotational symmetry" means in this context. A collection of random variables independently distributed according to identical "doubly-rectified" exponential distributions would also be symmetric under variable interchange.
However, this represents invariance only under sequences of axis-aligned rotations that are multiples of pi/2. The joint distribution of a collection of independently and identically normally-distributed random variables is invariant under arbitrary rotations, which is a much stronger form of invariance.
E.g., if you drew [x1...xn] from a distribution like exp(-|x1|-|x2|-...-|xn|), which is invariant with respect to variable interchange, and normalized to unit length, the resulting distribution would be markedly non-uniform over the surface of the n-dimensional unit hyper-sphere.
In fact, the differences in the symmetries of these distributions are crucial to the relative behaviors of L1 and L2 regularizers in machine learning. These differences have significant practical, and not merely theoretical consequences.
It is obvious if you think of it (exp(x1^2+...+xn^2) is symmetric on all the variables). Yep, understood.