How can i shape a Möbius Strip into another curve?

Question

The normal parametrisation of the Möbius strip:
$$\begin{align} x(u,v) &= (1+\frac{v}{2}\cos(u/2))\cos(u) \\ y(u,v) &= (1+\frac{v}{2}\cos(u/2))\sin(u)\\ z(u,v) &= \frac{v}{2}\sin(u/2)\end{align}\\ 0\leq u \leq 2\pi\: \text{and} -1\leq v\leq 1$$
gives me a Möbius stripe shaped in a circle form, I would like to have it shaped in the form of a different curve $g: [0,2\pi] \rightarrow \mathbb{R}^3$
Is it possible to do this? Where do I apply the parametrisation of my curve $g$? The naive approach to just apply the parametrisation of $g$ to the variable $u$ didn't go so well...