  In the field of topological insulators, the topological properties of quantum
states in samples with simple geometries, such as a cylinder or a ribbon, have
been classified and understood during the last decade. Here, we extend these
studies to a M\"obius band, and argue that its lack of orientability prevents a
smooth global definition of parity-odd quantities such as pseudovectors. In
particular, the Chern number, the topological invariant for the quantum Hall
effect, lies in this class. The definition of spin on the M\"obius band
translates into the idea of the orientable double cover, an analogy used to
explain the possibility of having the quantum spin Hall effect on the M\"obius
band. We also provide symmetry arguments to show the possible lattice
structures and Hamiltonian terms for which topological states may exist in a
M\"obius band, and we locate our systems in the classification of topological
states. Then, we propose a method to calculate M\"obius dispersions from those
of the cylinder, and we show the results for a honeycomb and a kagome M\"obius
band with different types of edge termination. Although the quantum spin Hall
effect may occur in these systems when intrinsic spin-orbit coupling is
present, the quantum Hall effect is more intricate and requires the presence of
a domain wall in the sample. We propose an experimental set-up which could
allow for the realization of the elusive quantum Hall effect in a M\"obius
band.
