Let $X$ be a $n$ by $n$ symmetric matrix. Let vec$(X)$ denote the operator that stacks the columns of $X$ into a row vector and let $\otimes$ denote the well-known Kronecker product.
The two $n^2$ by $n^2$ matrices vec$(X)$vec$(X)^{\top}$ and $X \otimes X$ contain the same elements but in a different order.
How are vec$(X)$vec$(X)^{\top}$ and $X \otimes X$ related algebraically?
EDIT: This is slightly more general and does not give an exact algebraic relation between the two.