  Neumann domains of Laplacian eigenfunctions form a natural counterpart of
nodal domains. The restriction of an eigenfunction to one of its nodal domains
is the first Dirichlet eigenfunction of that domain. This simple observation is
fundamental in many works on nodal domains. We consider a similar property for
Neumann domains. Namely, given a Laplacian eigenfunction $f$ and its Neumann
domain $\Omega$, what is the position of $\ .f\ |_{\Omega}$ in the Neumann
spectrum of $\Omega$?
  The current paper treats this spectral position problem on the
two-dimensional torus. We fully solve it for separable eigenfunctions on the
torus and complement our analytic solution with numerics for random waves on
the torus. These results answer questions from [37, 8] and arouse new ones.
