My answer isn’t a GQ quality answer, but maybe I can get the ball rolling, away from these interesting but otherwise pointless exceptions.
I’m nearly certain that the fact that they are “opposites” as you say is just coincidence. The digraph originated in the same place for both languages: from the Italian respelling of the Greek letter chi (X), then pronounced as an aspirated K (think a short whispered English “cut” without the T). Italian seems to have mostly lost its aspiration, so ch just became K. English, while it has not actually lost its aspiration, no longer considers it important, and also has evolved. But the way it did so is different–aspirated sounds became palated affricatives.
I don’t know why English uses the [s] sound before E and I, but it’s trivial to understand why Italian uses the sound it does. [e] and * are very front vowels, meaning they are articulated close to the soft palate. Thus it makes sense that the consonants before said sounds would be palated (produced with the soft palate). However, a palated stop (which causes all sound to stop) is difficult to make, so it makes sense to combine a palated fricative with a dental stop, which sounds really similar. And a fricative combined with a stop is what we call an affricative. The English T + SH = CH, and D + the French J (or the s in measure) = English J.