There are exactly two one plane bundles over the circle, the open annulus, and open Moebius band, where we take projection to be onto the central circle. We can now ask what the mod 2 self intersection of the zero section is. In the case of an annulus it is zero, in the case of the Moebius band it is one.
The tubular neighborhood theorem says that any smooth submanifold of a smooth manifold has a neighborhood homeomorphic to a k-plane bundle over the submanifold, where k is it's codimension.
You can't embed a Moebius band in an orientable surface, so the self intersection of any simple closed curve in an orientable surface is zero. The self intersection number is invariant under homotopy, so if gamma and gamma' are homotopic simple closed curves in a surface that intersect each other transversely then they intersect in an even number of points.