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?

If he has a friend then they both can paint their right hands blue and left hands red.
His friend stays where he is, he goes once around the strip, now his left hand and right hand are switched when he compares them to his friends hands.
