Except that isn't actually answering his question, you've just come up with some concept which allows you to count in whole numbers.
Besides, your method doesn't gel with the required behaviour. For example, fermionic spin is half integer spin, ie 1/2 or 3/2. Bosonic spins are 0, 1 or 2. Your method cannot build spins higher than 1, since you define it as 1/N if the number of twists are N. You can build spins of say 1/5, except no such spin exists.
You can view the spin as how the wavefunction changes, ie $$|psi\rangle \to e^{2\pi i s}|\psi \rangle$$ if $$|\psi\rangle$$ has spin s properties. If s is an integer than it doesn't change. If s is half integer it picks up a factor of -1. Anyons let s be anything except, as I said, they only arise in 2+1 dimensional space-time. You cannot have a Mobius strip in 2+1 dimensional space-time in the manner you describe so you cannot view them in such a manner.
Now there are topological things related to particles which can be viewed in terms of tori, Mobius strips, spheres and plenty more weird and wonderful spaces. However, the specific connection to these things is constructed formally. Many of them cannot be viewed geometrically because they are say 6 or 10 or 50 dimensional structures with multidimensional loops and handles and twists and other shapes we don't even have names for. Simply thinking "I need something which counts integers. Oh, a Mobius strip!" doesn't cut it. Personally if someone asked me to think of a space which counts integers I'd think a torus, since that's what its fundamental group measures. Twice!
For example, the reason you only get $$\pm 1$$ transformations in space-times with more than 2 spatial directions is to do with the first homotopy group of the space-time's metric symmetry group, which is $$\mathbb{Z}_{2}$$, which the group ( * , {1,-1} ) forms a representation of. If d=2 then you get $$\mathbb{Z}$$ and all sorts of problems arise. But if you didn't know any group theory and algebraic topology this would not be known. And this isn't mathematical navel gazing, 2+1 dimensional gauge theories are finding serious applications in the development of graphene based materials and condensed matter.
I can understand why you would find a simple geometric point of view palatable but you have absolutely no way of demonstrating such a point of view is actually any reflection of the behaviour of the system. The mathematical constructs in mainstream physics are derived from physically motivated premises and they allow specific predictions to be made and then tested. Approaches like Farsight's obsession with giving some simple geometric point of view are not going to be of any use if they are just plucked from nowhere. Given a specific property of a system it's easy to come up with another simple system which has a similar property. But if you have to give a different analogy for each property it's of no use because the analogies don't allow you to reason about anything else in the system. Unless there's some actual derivation/justification beyond the construction of a simple point of view there's absolutely no reason to think it's worth putting any trust in.
The application of mathematics to physics is precisely because we can trust the end result to be as sound as the premises. Unfortunately too many hacks try to delude themselves they can accomplish great things in developing physics models without actually formalising any models. When asked "What happens to a ball when it is thrown into the air" the answer "It comes down" is true but an answer which tells you where, when and how fast is the better one, no matter how palatable the justification for "It comes down" might seem.
