Hofstadter's book GEB talked about chess and intelligence a bit, and it mentioned some studies. There's one I most definitely remember. The study involved taking two groups of people, normal people and chess pros, showing them a chess board for like 5 seconds, and having them reconstruct as much of the board position as they could.
The normal people did exactly like you thought they would, placing a few pieces around, with a lot of off-by-one errors. The chess pros, on the other hand, got board positions comparatively "more wrong" than the beginners. Multiple pieces were radically moved across the board. However, when these board positions were looked at by other pros, the responses were almost universally along the lines of "well, the positions are different, but strategically they're actually kinda similar..."
It's qualitative feedback, to be sure, and difficult to verify and certainly not be trusted in a scientific sense. Nonetheless, it does reveal some of the "intuition" going on. Pro chess players are good at seeing high-level patterns in the positioning of the pieces, related to strategies and advantages and relative positioning. They see these things so well that they stop seeing the actual pieces, to an extent. It reminds me of that scene in the first Matrix movie where the one guy (Cypher?) is pointing to the screen with numbers trailing down, saying "I don't see the numbers anymore, it's just blonde, brunette, redhead, ..."
This seems to relate more to perception than to intuition as it's traditionally thought of, but I would argue that they two are more strongly coupled than most people realize. One type of intuition is perceiving emergent phenomena directly without noticing the underlying layers.