Is there a nice theorem about the algebra of invariants $\mathbb{C}[\mathfrak{gl}_n]^{SO_n \times SO_n}$, where the action is by left and right multiplication? I'm hoping for something along the lines of the Chevalley restriction theorem, with a list of generators for the algebra being a nice bonus if possible (the subalgebra should have dimension $n$; I'm hoping it's free with $n$ generators).

For example, for $n = 2$, the algebra of invariants is generated by the determinant and the sum of squares of components.

My slightly-educated guess is that the theorem should be approximately:

$\mathbb{C}[\mathfrak{gl}_n]^{O_n \times O_n} \simeq \mathbb{C}[\mathfrak{h}]^{W'}$, where $W' = W \rtimes C_2^n$, $C_2$ is the cyclic group of order 2, the action of $W = S_n$ on $C_2^n$ comes from permutation, and the action of $C_2^n$ on $\mathfrak{h}$ is by negation of a coordinate. Generators for the latter algebra would be symmetric polynomials on the squares of coordinates in $\mathfrak{h}$, with degrees $2, 4, 6, ... 2n$. Then $\mathbb{C}[\mathfrak{gl}_n]^{SO_n \times SO_n} = \mathbb{C}[\mathfrak{gl}_n]^{O_n \times O_n}[\text{det}]$; generators would be the same, except the generator of highest degree (degree $2n$) would be replaced with the determinant.