  Let $\mu_2(\Omega)$ be the first positive eigenvalue of the Neumann Laplacian
in a bounded domain $\Omega \subset \mathbb{R}^N$. It was proved by Szego for
$N=2$ and by Weinberger for $N \geq 2$ that among all equimeasurable domains
$\mu_2(\Omega)$ attains its global maximum if $\Omega$ is a ball. In the
present work, we develop the approach of Weinberger in two directions. Firstly,
we refine the Szego-Weinberger result for a class of domains of the form
$\Omega_{\text{out}} \setminus \overline{\Omega}_{\text{in}}$ which are either
centrally symmetric or symmetric of order $2$ (with respect to any coordinate
plane $(x_i,x_j)$) by showing that $\mu_{2}(\Omega_{\text{out}} \setminus
\overline{\Omega}_{\text{in}}) \leq \mu_2(B_\beta \setminus
\overline{B}_\alpha)$, where $B_\alpha, B_\beta$ are balls centered at the
origin such that $B_\alpha \subset \Omega_{\text{in}}$ and
$|\Omega_{\text{out}}\setminus \overline{\Omega}_{\text{in}}|=|B_\beta
\setminus \overline{B}_\alpha|$. Secondly, we provide Szeg\H{o}-Weinberger type
inequalities for higher eigenvalues by imposing additional symmetry assumptions
on the domain. Namely, if $\Omega_{\text{out}} \setminus
\overline{\Omega}_{\text{in}}$ is symmetric of order $4$, then we prove
$\mu_{i}(\Omega_{\text{out}} \setminus \overline{\Omega}_{\text{in}}) \leq
\mu_i(B_\beta \setminus \overline{B}_\alpha)$ for $i=3,\dots,N+2$, where we
also allow $\Omega_{\text{in}}$ and $B_\alpha$ to be empty. If $N=2$ and the
domain is symmetric of order $8$, then the latter inequality persists for
$i=5$. Counterexamples to the obtained inequalities for domains outside of the
considered symmetry classes are given. The existence and properties of
nonradial domains with required symmetries in higher dimensions are discussed.
As an auxiliary result, we obtain the non-radiality of the eigenfunctions
associated to $\mu_{N+2}(B_\beta \setminus \overline{B}_\alpha)$.
