I just thought of something else that you might consider. I remember reading a review years ago by JA of Stereophile where he discovered a pair of PSB speakers that were wired backwards. The woofer binding posts were driving the tweeters while the tweeter binding posts were driving the woofers. Depending on whether or not the Triangles have dual binding posts and how you are cabling them, I suppose this could alter the sound characteristics.
However, the fact that you have swapped tweeters and noticed an improvement has me a bit baffled. If a crossover has a problem, I wouldn't expect a change from swapping tweeters. Additionally, if a tweeter has a problem, I would expect it to continue to have a problem once moved to the other speaker.
I'm wondering if perhaps the tweeter was wired in a reverse polarity inside the cabinet and corrected inadvertently by you when you swapped them out. This would certainly account for an improvement and your perception that the sound seems less pronounced might simply be their natural voicing characteristics that were being masked.