For example, consider the Laplacian on $[-1,1] \times [-1,1]$ with Dirichlet boundary conditions. If $m$ and $n$ are distinct positive integers, $\sin(m \pi x) \sin(n \pi y)$ and $\sin(n \pi x) \sin(m \pi y)$ are both eigenfunctions for the same eigenvalue, and are both odd. Similarly $\cos((m+1/2) \pi x) \cos((n+1/2) \pi y)$ and $\cos((n+1/2) \pi x) \cos((m+1/2) \pi y)$ which are both even.

For example, consider the Laplacian on $[-1,1] \times [-1,1]$ with Dirichlet boundary conditions. If $m$ and $n$ are distinct positive integers, $\sin(m \pi x) \sin(n \pi y)$ and $\sin(n \pi x) \sin(m \pi y)$ are both eigenfunctions for the same eigenvalue, and are both odd. Similarly $\cos((m+1/2) \pi x) \cos((n+1/2) \pi y)$ and $\cos((n+1/2) \pi x) \cos((m+1/2) \pi y)$ which are both even.

EDIT: Cases where a positive integer can be represented in many ways as a sum of squares present even more multiplicity.