OK, the problem was the MyTube module (1.05RC).
This was the way I got there:
1) first, I checked the permissions on xoops_data, but all of them were 777, so it was good
2) I checked for any D3 modules (e.g. Protector or AltSys), because if the xoops_lib path is messed up, that causes a problem as well, but everything was good there
3) As next we need to check the modules. Normally if there is a problem with an unistalled module, then the guilty one is the one that is missing as first one alphabetically. But at least all the installed should show up.
In your case only the System was visible in the center, so it was weird, because there were other modules installed (cbb, statistics, etc.), but they were visible only on the left hand side, but not in the center.
So the next step was to delete one by one the "unistalled" modules and see if there is a problem with any of them. The first one was dictionary, but it was fine. The next one was "MyTube", and as soon as deleted it, all the modules were visible in the center.
Now comes the explanation why all the other installed modules were not visible. It was because you've installed MyTube, but deactivated it. So it was not visible on the left hand side because it was not active, but because it was the first one alphabetically as installed, then it caused all the following ones to become invisible.
I've copied back the Dictionary and MyTube folders, but changed the name of the MyTube folder, so XOOPS thinks that it is missing and thus it is able to show it in the center.
The problem is that it's not possible to uninstall MyTube automatically, and I don't know why, therefore you'll need to do it manually via phpMyAdmin.
For next time, you need to keep notice what steps do you take and at what moment things don't work out anymore.
From the modifications dates it looks like you've installed XOOPS 2.3.3 on March 17th. Then on April 8th you've installed the Statistics module and recompiled the other modules. But the MyTube was recompiled on April 10th. So this should tell you that since MyTube was the last one module that you've installed, the problem could it MyTube.
Anyway, the mystery is solved