In a 1D interval $[a, b]$, quadrature rules and orthogonal polynomials are tightly interconnected. For example, given the $n$ roots $t_j$ of the $n$th orthogonal polynomial $p_n$, one has the following classical theorem:
Given $f\in C^{2N}([a, b])$, $$ \int_a^b f(x) \,\text{d}x = \sum_{j=1}^N w_j f(t_j) + \frac{f^{(2n)}(\xi)}{(2N)!k_N^2} \quad (a<\xi<b) $$ (where there weights $w_j$ can be computed accordingly).
In other words: There is a quadrature rule that is exact for polynomials of degree up to $2N-1$ with the quadrature points being the roots of the $n$th orthogonal polynomial.
If you look at higher-dimensional domains, e.g., triangles, series of orthogonal polynomials are known. (Check out orthopy, for example, a small project of mine.)
Question: Is there a statement similar to the above that connects properties (roots?) of the orthogonal polynomials to quadrature rules?