Consider the Möbius strip as the unit square with two opposite sides identified (with opposite directions).   Consider the eigenvalue equation $\Delta u = \lambda u$ with boundary condition $u=0$.  Unlike for orientable manifolds, the least eigenfunction will not be all of one sign;  there will be a nodal line.  My question generally concerns the behavior of eigenfunctions and eigenvalues in the non-orientable case, but to ask some specific questions:  (1) is the eigenspace of the first eigenvalue still one-dimensional?  (2) does there have to be just ONE nodal line?  (3) does any nodal line have to meet the boundary in two points?