Thanks for your reply.
It's exactly as you describe. However, input methods seem to depend on the application. Within the terminal (where I played around in the first place) for instance it is not possible to switch to Katakana mode. Hiragana mode is possible but ignored, i.e. input is still Romaji.
But what about "kanji"? I would expect that button to fire up the CRIM. Well then, I installed other input method which might have overwritten the default behaviour? If so, how to reset it?