  We consider the dynamics of a two-dimensional incompressible perfect fluid on
a M\"obius strip embedded in $\mathbb{R}^3$. The vorticity-streamfunction
formulation of the Euler equations is derived from an exterior-calculus form of
the momentum equation. The non-orientability of the M\"obius strip and the
distinction between forms and pseudo-forms this introduces lead to unusual
properties: a boundary condition is provided by the conservation of circulation
along the single boundary of the strip, and there is no integral conservation
for the vorticity or for any odd function thereof. A finite-difference
numerical implementation is used to illustrate the M\"obius-strip realisation
of familiar phenomena: translation of vortices along boundaries, shear
instability, and decaying turbulence.
