You can also see that in a case where there is a contradiction between logic and intuition, the very person who says there is a contradiction initially chose logic, because that person chose to determine that there is a contradiction and that he must cast one thing aside, rather than determine that there is no contradiction in some non-logical way.
It also seems possible to say that there is a hierarchy among intuitions according to their degree of clarity and coherence, and that the intuition on which logic stands is the most basic of all. For the intuition on which logic stands is an intuition without which it is impossible to create consistent thoughts, and inconsistent thoughts have no intellectual relevance for us.

I agree with the first remark. As for the second, I agree that there is a hierarchy among intuitions (and therefore the question is not well defined, because it assumes that intuition has some given validity). But for some time now I have been wondering whether logic really is based on intuition, because in my view it asserts nothing, and therefore there is no need to ground its claims in anything. It simply points to relations between things, and they are such because they are such. No foundation is needed for that. It seems to me that I even once wrote that intuition is the foundation of logic, but today I am inclined not to think so.

1. I will begin with your last sentence:
"Intuition is the foundation of logic."

2. This sentence expresses exactly my claim.

3. The intuitive faculty is, in my opinion, the "image of God" in man. Insofar as one may say that God is endowed with understanding and knowledge, His knowledge is full and immediate and does not depend on the axis of time.
In the same way, the intuitive faculty itself (which He implanted in us) is full, immediate, and "supra-temporal."

4. By contrast, logic expresses the human side of the human spirit insofar as it has a discursive character: it builds its object piece by piece (from premise to conclusion), it does so in a mediated way, and at least from the human perspective it moves from potentiality to actuality within the axis of time.

5. In short: logic attains its object (say, when it creates a definition or a proposition) as a pale "imitation" of the way intuition does so.

6. The intuitive faculty was granted to us by God so that we could flirt a little with metaphysical truths (with Him Himself standing at their center).

7. In this respect logic is made possible by the intuitive faculty (and perhaps even derives from it).

8. From the fact that we decide, as you say, by means of logic between contradictory intuitions, it does not follow that logic is more "necessary."

9. The reason is that my question was not about specific cases of intuitive cognition, but about the primary sources of the human spirit (within which both intuitions and logic are found).

It seems to me that by now we have reached the final stage. Each side has said its piece, and the chooser may choose.

That was quick. Not really reasoned or substantive, but definitely quick. With a bit of effort on my part I can imagine possible worlds in which a philosophical criterion of this sort would serve its owner.