I was attempting to describe the inside out motion that is realised by turning a Moebius band which has no self intersections.The Moebius band considered would be a ruled surface parametrised as
$$\boldsymbol\sigma(t,\theta)=\left(\left(1-t\sin\dfrac\theta2\right)\cos\theta,\left(1-t\sin\dfrac\theta2\right)\sin\theta,t\cos\dfrac\theta2\right),$$ where the domain of definition of $\boldsymbol\sigma$ is $$U=\{(t,\theta), \mid-1/2<t<1/2,\ 0<\theta<2\pi\}.$$
However I am unsure whether the right way is to model this phenomenon as a family of isometries or through time evolution of the ruling vector taken as a hinge and rotating about the midline? I found a paper on inverting a cylinder by Halpern and Weaver.