  In this paper we study a number of conjectures on the behavior of the value
distribution of eigenfunctions. On the two dimensional torus we observe that
the symmetry conjecture holds in the strongest possible sense. On the other
hand we provide a counterexample for higher dimensional tori, which relies on a
computer assisted argument. Moreover we prove a theorem on the distribution
symmetry of a certain class of trigonometric polynomials that might be of
independent interest.
