Can a 2D person walking on a Möbius strip prove that it's on a Möbius strip? Or other non-orientable surface, can a 2D walker on a non-orientable surface prove that the surface is non-orientable or does it always take an observer from a next dimension to prove that an entity of a lower dimension is non-orientable? So it always takes a next dimension to prove that an entity of the current dimension is non-orientable?

Flatlanders of Relativity theory as we know can measure motion with successive motion derivatives to get information about spatial orientation on an inertial frame. 
Torus/Möbius Band homeomorphism 
A slender Möbius band is very different from a bulky Möbius band of rectangular or near square section, please note. You get  orientability  and navigation without external reference  when you dare to look and go sideways around its edge or cross the border under your feet through a hole in the band taking a short cut.  
EDIT1:
In the link below, I observed that as the Möbius Band gets fatter, it gains in orientability due to increased access though or across its thickness. 
Torus/Möbius Band homeomorphism
One take away view from there is that the Möbius band is very different from the fattened Möbius band; the non-orientability of the former doesn’t imply anything about the latter.
If inertial navigation is possible by local measurements made by FlatLanders, imho, it removes sense from the notion of orientability of the Möbius Band.
