I still haven't seen any examples that aren't -ics words (besides 'news'), but let me post a tentative answer for the -ics case: I remembered something I had quoted in the inning/innings singular/plural question, and looked up Fowler. Under -ics, the book says:
-ics. 1. Among the names of sciences, arts, or branches of study, are a few words in -ic that rank as real English; the chief are logic, magic, music, & rhetoric; but the normal form is -ics, as in acoustics, classics, dynamics, ethics, mathematics, physics, politics, tactics. The substitution of -ic for -ics …
2. Grammatical number of -ics. This is not so simple a matter as it is sometimes thought. The natural tendency is to start with a fallacy: We say Mathematics is (& not are) a science; therefore mathematics is singular. But the number of is there is at least influenced, if not (whether legitimately or otherwise) determined, by that of a science. The testing should be done with sentences in which there is not a noun complement to confuse the issue:—Classics are, or is, now taking a back seat ; Conics is, or are, easier than I expected ; What are, or is, his mathematics like? ; Politics are, or is, most fascinating ; Your heroics are, or is, wasted on me ; Athletics are, or is, rampant in the big schools ; Tactics are, or is, subordinate to strategy. The rules that emerge are: (1) Singular for the name of a science strictly so used; Metaphysics, or Acoustics, deals with abstractions, or sound. (2) Plural for those same names more loosely used, e.g. for a manifestation of qualities; often recognizable by the presence of his, the, &c.: His mathematics are weak ; Such ethics are abominable ; The acoustics of the hall are faulty. (3) Plural for names denoting courses of action or the like : Heroics are out of place ; Hysterics leave me cold. (4) The presence of a singular noun complement often makes the verb singular: Mathematics, or even Athletics, is his strong point.
So that long discussion very well illustrates the circumstances in which -ics words are sometimes used as singular and sometimes plural (though I wouldn't say "His mathematics are weak", for some reason). Note that it seems to be discussing when a certain -ics word is singular and when it's plural, not saying that it's always plural and discussing when it's treated one way or the other.
Anyway, along with RegDwight's useful answer, my hypothesis for my original question is this: if a singular word X exists (e.g. mathematic), and the word Y is formed from X by the normal rules of English pluralisation, then word Y is called "plural", even if it has never been used in the plural.
Of course, all this still doesn't answer the latter question of what purpose this category serves etc.