  Let $\mathfrak{g}=\mathfrak{g}_0\oplus \mathfrak{g}_1$ be a $\mathbb
Z_2$-grading of a classical Lie algebra such that $(\mathfrak{g},
\mathfrak{g}_0)$ is a classical symmetric pair. Let $G$ be a classical group
with Lie algebra $\mathfrak{g}$ and let $G_0$ be the connected subgroup of $G$
with ${\rm Lie} (G_0)=\mathfrak g_0$. For $d \geq 2$, let
$\mathfrak{C}^d(\mathfrak{g}_1)$ be the $d$-th commuting scheme associated with
the symmetric pair $(\mathfrak g, \mathfrak g_0)$. In this article, we study
the categorical quotient $\mathfrak{C}^d(\mathfrak{g}_1)//{G_0}$ via the
Chevalley restriction map. As a consequence we show that the categorical
quotient scheme $\mathfrak C^d(\mathfrak g_1)//G_0$ is normal and reduced. As a
part of the proof, we describe a generating set for the algebra
$k[\mathfrak{g}_1^d]^{G_0}$, which are of independent interest.
