was soll das Wort denn ausdrücken?

"Counter[-]intuitiveness" (with or without a hyphen) sounds more plausible and registers a lot more Google hits, but I have not found it in any dictionary (just used my intuition...). "Intuitiveness" is definitely a word, so this should be ok.

"Counter[-]intuition", on the other hand, appears in some dictionaries but perhaps isn't quite what you're trying to express.

@.: the degree to which something is counterintuitive, of course! :-)

Das Adjektiv gibt's auf jeden Fall.
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I am a bit puzzled about the word "counterintuition". What is that supposed to mean? I could think of adjective-like use in compounds such as "the counter-intuition behaviour of the system" (where it would mean the very same as "counterintuitive"); however American Heritage, the only dictionary I could find it in, gives the following definition: "Intuition that is contrary to what one would expect."
How can intuition be contrary to what one would expect? Sounds like an oxymoron to me. Can anyone think of a reasonable context where this would fit? I.e. a context likely enough to deserve an own word, not something exotic that is made-up just for it?

"counterintuitive" is standard (though somewhat educated) English. But other presumed cognates, "counterintuitiveness" "counterintuivity" etc., just sound clumsy and bad. Even if you could find them being used somewhere, I wouldn't imitate it. It sounds very jargony and unnatural, like bad corporate or academic writing.

My two cents.

I would disagree with counterboy in one particular: To my mind, "counterintuitiveness" is perfectly acceptable if used in the right context (where the quality or extent of something's being counterintuitive is at issue). Of course, it is not an everyday sort of word, and that does make it "jargony" in the neutral sense of that word ("pertaining to a specific specialist field of discourse"), but I see no reason to consider it either "jargony" in the negative sense ("producing obfuscation by means of technical or pseudo-technical language") or "bad" (whatever counterboy meant by that).

For me, the first test as to whether a word is worthy of being struck is whether it carries a useful meaning. The meaning of counter-intuitiveness is absolutely clear. Then, I consider whether a different word or expression may carry the same meaning in a more elegant or efficient way (according to whatever criteria apply to the text/situation in question).

A blanket dismissal of "counterintuitiveness" would have to apply to all contexts, times and places, which is clearly absurd.

To summarize my view: For writers with a dry, technical, polysyllabic and latinate style, I would recommend this word as a handy addition to an advanced vocabulary. If you feel it is right, use it, for there is nothing wrong with the word itself. For politicians, advertisers, tabloid journalists and others who have to communicate with the general public on a regular basis (and whose ability to put food on their families' tables may depend on the success of this communicative enterprise), this word, though eminently comprehensible, is clearly too long, complex and precise to waste on your audiences and shoue be replaced with several shorter (four-letter?) words that will carry a vaguely reassuring feel and not encourage people to think (do not forget: thought = lost support/revenue/readers).