I apologise for the vague statement in the title.
A fiber bundle is often viewed as a smooth family of fibers indexed smoothly by points of a manifold (called the base space). I am imagining the fibers vertically (like a line segment for a mobius strip). In this viewpoint, the sections of the bundle are horizontal cuts.
I would expect that there is an equivalent viewpoint that interchanges horizontal and vertical roles. Suppose, I would like to begin with foliations (of codimension 1?) on a manifold (that play the role of distinguished sections). Is there a way to construct extra structure that recovers the notion of a fibre and prove equivalence with the notion of a fibre bundle?