Pass to the double cover, which is a cylinder: $[0,1] \times [0,2]$ with the end points of the second interval identified. On the cylinder, the eigenfunctions are $f(x,y) = \sin (ax \pi) e^{i b x y \pi}$ with $a$ and $b$ integers and $\lambda = - a^2 - b^2$. An eigenfunction descends to the Mobius strip if and only if $f(x,y) = f(-xf(1-x, y+1)$ which means that $b$ is odd.
Pass to the double cover, which is a cylinder: $[0,1] \times [0,2]$ with the end points of the second interval identified. On the cylinder, the eigenfunctions are $f(x,y) = \sin (ax \pi) e^{i b x \pi}$ with $a$ and $b$ integers and $\lambda = - a^2 - b^2$. An eigenfunction descends to the Mobius strip if and only if $f(x,y) = f(-x, y+1)$ which means that $b$ is odd.