PROBLEM SOLVED... but...
Well, I solved the problem. It was really easy. I just had to connect a second integrated amplifier that only powers the tweeters. Then, reduce the tweeters' sound by 30-40%. Now, my speakers can face at me again and all sounds very normal and authentic.
But why why why? - Here is what I have been through in the meantime, and I am still puzzled:
1) I bought myself an ultrasonic dog whistle with adjustable pitch. When I adjust it to the highest pitch, I can not only still hear it, but it is so loud that it is ear-piercing. Others either don't hear it at all, or hear it as a very soft sound.
2) I consulted an audiologist and he confirmed that my hearing is excellent... in fact, beyond his equipment measuring capabilities. My audiogram is almost flat, with a small peak at 2,000Hz. BTW the audiologist could not hear the dog whistle either...
3) I bought myself a
calibration CD. Now, that was really interesting when it came to the various frequencies samples:
a) With the default settings (whether using my Sony solid state, or a tube amp) I can hear all frequencies between 22Hz (earthquake) and 10,000Hz (not quite as high pitch as the dog whistle), and I hear them all at the same volume. Below and beyond that, I do not hear anything at all.
b) With my preferred settings (with tweeters' sound reduced by 30-40%), I can hear all frequencies between 22Hz and 8,000Hz, and I hear them all at the same volume. Below and beyond that, I do not hear anything at all.
So what does that mean???
My speakers (I tested Heybrook and Tannoy), and the rest of my equipment (whether solid state or tube), are supposed to reproduce sounds up to 20,000Hz... but that is definitely not true, since the highest I can hear from them is 10,000Hz (and that is still a lower pitch than my dog whistle, which is ear-piercing to me).
So, if my equipment does not reproduce enough treble, I should hear not enough treble, right? - Wrong! - I hear too much of it, and I have to turn it down!
Anybody understands that mystery?
It is something to do with my audio system (or rather, with audio systems in general), since listening to live classical music never poses me any problems.