Integration over a Cartesian product

Question

Let $x_1,x_2\in\mathbb{R}^d$ and $p(x_1,x_2):\mathbb{R}^{d}\times\mathbb{R}^d\mapsto\mathbb{R}$ be a generic scalar function. Let $S\subseteq\mathbb{R}^d$ be a generic domain of integration. Assume that the following integral exists \begin{equation*} \int_{S^2} p(x_1,x_2)\text{ d}x_1\text{d}x_2 \end{equation*} where $S^2\triangleq S\times S$. Is it true the following equality? \begin{equation*} \int_{S^2} p(x_2,x_1)\text{ d}x_1\text{d}x_2 = \int_{S^2} p(x_1,x_2)\text{ d}x_1\text{d}x_2 \end{equation*} I believe that the answer is yes because, if I'm not wrong, the integration domain $S^2$ is symmetric with respect to $x_1,x_2$, in the following sense: if we have a parametrization $\phi_{S^2}(t)=[x_1(t)\,\,x_2(t)]'$ to represent $S^2$, then even the permutated parametrization $\tilde{\phi}_{S^2}(t)=[x_2(t)\,\,x_1(t)]'$ should be able to identify the points forming $S^2$.