Suppose we have a matrix $A$ with dimensions $m$ by $n$ and a column-wise permutation matrix $R$ (re-orders columns) with dimensions $n$ by $n$.
Then we have a matrix $X$ which is constructed as
$X = A (A R)^T$
$X = A R^T A$
Let the SVD of $X = U_X S_X V_X^T$.
Then since
$A = U_A S_A V_A^T$
we have
$X = U_A S_A V_A^T R^T V_A S_A^T U_A^T$
is it possible to obtain the SVD of $X$ ($U_X, S_X$ and $V_X$) or even just the left and right singular vectors using the SVD decomposition of $A$?