  We investigate multiplicity and symmetry properties of higher eigenvalues and
eigenfunctions of the $p$-Laplacian under homogeneous Dirichlet boundary
conditions on certain symmetric domains $\Omega \subset \mathbb{R}^N$. By means
of topological arguments, we show how symmetries of $\Omega$ help to construct
subsets of $W_0^{1,p}(\Omega)$ with suitably high Krasnosel'ski\u{\i} genus. In
particular, if $\Omega$ is a ball $B \subset \mathbb{R}^N$, we obtain the
following chain of inequalities: $$ \lambda_2(p;B) \leq \dots \leq
\lambda_{N+1}(p;B) \leq \lambda_\ominus(p;B). $$ Here $\lambda_i(p;B)$ are
variational eigenvalues of the $p$-Laplacian on $B$, and $\lambda_\ominus(p;B)$
is the eigenvalue which has an associated eigenfunction whose nodal set is an
equatorial section of $B$. If $\lambda_2(p;B)=\lambda_\ominus(p;B)$, as it
holds true for $p=2$, the result implies that the multiplicity of the second
eigenvalue is at least $N$. In the case $N=2$, we can deduce that any third
eigenfunction of the $p$-Laplacian on a disc is nonradial. The case of other
symmetric domains and the limit cases $p=1$, $p=\infty$ are also considered.
