Let us consider gluing many patches to obtain a nonorientable manifold.
If we have an intersection of 3 patches, the transition functions must be consistent on this intersection, i.e. there is some thing known as a cocycle condition.
Moreover, some of patch changes are now orientation-reversing.
We need to define and specify transition functions and orientation-reversing patches on how they behave in a nonorientable manifold.
I am asking the principle behinds:
What are the rules of transition functions and orientation-reversing patches on a nonorientable manifold:
(1) for complex line bundles? [thus we only require the SO(n) structure and O(n) structure for manifolds. ]
(2) for the spinor? [thus we require the Spin structure and Pin structure for manifolds.]