Rotational symmetries of domains and orthogonality relations

Let $\mathrm{\Omega }\subset {\mathbb{C}}^n$ be a domain whose Bergman space contains all holomorphic monomials. We derive sufficient conditions for $\mathrm{\Omega }$ to be Reinhardt, complete Reinhardt, circular or Hartogs in terms of the orthogonality relations of the monomials with respect to their $L^2$-inner products and their $L^2$-norms. More generally, we give sufficient conditions for $\mathrm{\Omega }$ to be invariant under a linear group action of an $r$-dimensional torus, where $r\in \{1,\dots ,n\}$.