Another typographical convention that intuits the similarity is the rule that when you break a quotation into paragraphs, you open each paragraph with inverted commas, but only put the reverted commas at the end of the entire quotation. (Cf. Virginia Woolf's The Waves as a good example of the Hogarth Press's conformity to this rule.) Similarly, parentheticals that are broken into paragraphs have opening (concave) lunulae at the beginning of every paragraph but closing (convex) lunulae only at the end of the entire parenthesis (I am using “concave” and “convex” as understood intuitively, perhaps: the opening lunula opens an interior space: the closing lunula pushes us onward into the flow of the larger discourse).
I was thinking about this the other day, and realizing that there is an interesting and symmetrical difference between quotations and parentheses. A parenthetical phrase (like this one) may refer to things outside of it, parts of the sentence it inhabits (say) that have no reciprocal need for the parenthesis (which is why it's parenthetical; look at how cleverly Pope allows you so skip parentheses in The Rape of the Lock without disturbing the rhyme scheme (though parenthetical phrases will often contribute (“(not in vain)” (The Essay on Criticism)) to the meter)).
Quotations on the other hand must not refer to the quoting context, since they precede it logically and temporally. (“Scare quoted” material may, I suppose, but here they're pretty much meant to quote the context.) So parentheses are outward-looking, supplemental to the discourse in which they appear, but quotation is inward-looking. The quoting context is the late-coming supplement, unregarded by the haughty indifference of the quoted words.
This work previously appeared in October 2019 at http://www.balaustion.com.“Parentheses” (and quotations).