Thanks for checking back, Xavier.
From your response I can tell it should work, that's good news to start with. Now let's see where I screwed? Thanks in advance for hanging in with me on the subject.
First you must know that, to exclude the possibility of my customization causing it, as usual in such cases I set HikaShop back to 'out of the box' by removing all folders in my_template/html/com_hikashop, as well as replacing relevant modified helpers or classes files with their originals. Sorry, I should have maybe mentioned this. Anyway, always taking those steps first to cause you as little work as possible 
However, no Compares also without customizations.
Let me show you the menu/module configs and try to explain:
Under 'product category' I need 3 levels deep of product categories. So, on first level there are 4 what I call 'main categories', then each has a 2nd level of sub-categories, and some of those have then a 3rd level of sub-sub-categories.
Comparing is interesting really only on 2nd and 3rd level; which is where it doesn't work, unfortunately.
Just FYI, there's only one menu calling all products across all main categories (and no comparing wanted/needed), for the rest I'm staying within the main categories, so that a limitation to 2 levels is not a problem, and we here can focus on one of my main categories (named "Audio") - the other 3 are set up identically except for the associated category they're pulling from.
"Audio" has 14 categories under it, and there are up to 4 categories under some of those.
Please see attached a screenshot of the menu settings incl the associated module.
This menu produces correctly
(a) a menu/list of all (sub-)categories on 2 levels under my main category "Audio", and
(b) all products from all (sub-)categories under "Audio",
(c) with Compare
As soon as I click on any of the (sub-)category links, no matter which level, I'm getting
(a) eventually the correct menu/list of sub-categories - provided there are any, and
(b) the correct products in these sub-categories only,
but - incorrectly - no Compare
So, the pattern is: Compare = yes on top category level, Compare = no on lower category levels.
Again, any customization removed, no CSS hiding Compare.
How in the world did I screw this up?