  We introduce an analogue of Payne's nodal line conjecture, which asserts that
the nodal (zero) set of any eigenfunction associated with the second eigenvalue
of the Dirichlet Laplacian on a bounded planar domain should reach the boundary
of the domain. The assertion here is that any eigenfunction associated with the
first nontrivial eigenvalue of the Neumann Laplacian on a domain $\Omega$ with
rotational symmetry of order two (i.e., $x\in\Omega$ iff $-x\in\Omega$) "should
normally" be rotationally antisymmetric. We give both positive and negative
results which highlight the heuristic similarity of this assertion to the nodal
line conjecture, while demonstrating that the extra structure of the problem
makes it easier to obtain stronger statements: it is true for all simply
connected planar domains, while there is a counterexample domain homeomorphic
to a disk with two holes.
