Suppose I draw from the Gaussian orthogonal ensemble an $N\times N$ matrix $\mathbf{H}$ with eigenvalues $\lambda_1, \ldots, \lambda_N$. If I condition these $\mathbf{H}$'s so that $\lambda_1 \in [L_1 + dL_1], \ldots, \lambda_N \in [L_N + dL_N]$, can I expect the associated diagonalizing orthogonal transformations $\mathbf{O}$ to be Haar distributed? Would things change if the real, symmetric $\mathbf{H}$'s were distributed according to a different random matrix ensemble?
EDIT: Oh wait, this is trivial because the GOE is orthogonally invariant... I guess this is kind of neat because this allows a way to sample from other orthogonally invariant distributions given a known joint pdf for the eigenvalues?