I would suggest looking up the word "correspond" in a dictionary to start with. It has another meaning which has nothing to do with letter-writing. "Corresponding" is a more specific relationship than "appropriateness", and so could be a better choice in your cases.
Is there any significant difference between Correspond to and Correspond with? I only mean in the sense of "matching", here, rather than "communication". I've looked at a few sources, but I can't...
Correspond to vs. Correspond with - English Language & Usage Stack Exchange
6 In American English, in words ending with -age, -ate and -ace, the ‹a› correspond to /ɪ/ (short i). Examples: image, village, damage private, senate, separate surface, preface, palace (It should be noted that dictionaries do not always agree about the pronunciation, and some use /ə/ instead of /ɪ/ for some of the words).
Why does the letter "a" correspond to /ɪ/ in words like "image ...
Thanks for the comment. But in your two examples, "I visited Paris and Rome respectively" and "Each one visited her own mother respectively," "respectively" is definitely not needed. In your first example, there is nothing for Paris and Rome to correspond to, so "respectively" does not relay any information.
They instead more closely correspond to pairs of correlative conjunctions like As Y, so Y as in As goes the south, so goes the nation. Whether words like these should be assigned a classical part of speech of adverb or preposition or conjunction varies by analysis model.
3 mismatch a failure to correspond or match; a discrepancy Oxford English Dictionary This word might have the disadvantage of focussing more on the reasons to not fit in, "not-like-the-others" because of fashion, ideology, or some other superficial reason.