  The question of determining for which eigenvalues there exists an
eigenfunction which has the same number of nodal domains as the label of the
associated eigenvalue (Courant-sharp property) was motivated by the analysis of
minimal spectral partitions. In previous works, many examples have been
analyzed corresponding to squares, rectangles, disks, triangles, tori, \ldots .
A natural toy model for further investigations is the M\"obius strip, a
non-orientable surface with Euler characteristic $0$, and particularly the
"square" M\"obius strip whose eigenvalues have higher multiplicities. In this
case, we prove that the only Courant-sharp Dirichlet eigenvalues are the first
and the second, and we exhibit peculiar nodal patterns.
