Clear back to the sickel mower days of the 60's we doubled up the windrows. We would go in a circle skipping a rake pass on every round. When we got to the center we would turn around and throw the skipped pass into the windrow going the opposite direction.
Always did it this way with the mower conditioner as well.
Guess we were wrong for 50 years . But it worked fine without leaving any hay.
Maybe if the rake is junk it helps to go the same direction but we went any way with the rake, or baler for that matter, and never thought about.