In the QR-decomposition of a matrix $\mathbf{U}^T$ with pivoting, the permutation matrix can be expressed as a vector that re-orders the columns of the matrix in a manner such that the diagonal elements of $\mathbf{R}$ are non-increasing.
If $\mathbf{U}$ is the left unitary matrix in a singular-value decomposition of $\mathbf{M}^T$, what information about the significance of the columns of $\mathbf{M}$ does this ordering provide?
None. You will never pivot when you compute a QR factorization of a matrix which is already unitary. Hence the permutation matrix is always the identity matrix.