Let $W = X^TX$ denote a standard Wishart matrix, i.e., where $X$ is a Gaussian random matrix with iid standard Normal entries.

In this case we can write $W = U D U^T$, where $U$ is orthogonal and $D$ is the diagonal matrix of nonnegative eigenvalues.

My question is: are $(U, D)$ independent? If so, would there be a place that would contain proofs of such facts?

Of course, this is obvious in the case that $X$ is a single Gaussian vector, as then $D$ corresponds to the squared norm and $U$ corresponds to the direction, which are well known to be independent.