Finally after many tests I decided to start from scratch again.
I setup a new installation and copied the contect from the old installation (except the permissions, that were re-set manually).
Now it works perfectly.
So, permissions were all messed up and prevented TW to work as expected.
How we did get to this messy state? No clue
I attach one SQL file we used to transfer content from old messy DB to a new one, might it help someone.
Many thanks to Torsten and Bernard!!!