How does one parameterize the surface formed by a *real paper* Möbius strip?

The Möbius strip you show is a developable surface. No one, as far as I know, has been able to create a parametrization of it.

Since 1858, when the Möbius strip was discovered, mathematicians have been looking for a way to model it. The problem was finally solved in 2007 by E.L. Starostin and G.H.M. van der Heijden.

You might want to read their paper "The equilibrium shape of an elastic developable Möbius strip" by going to this site - http://www.ucl.ac.uk/~ucesgvd/pamm.pdf