On the rare occasion that it comes up, I’ve always called them “sexed” quotes, but it seems I get a lot of funny looks when I do. So I turned to the oracle and found little to defend that use. Wikipedia’s article refers to them variously as “book quotes,” “curved quotes,” “curly quotes,” and, presumably thanks to Microsoft Word, “smart quotes.”
That’s not to say that there are no references to “sexed quotes.” That phrase draws 48 hits from Google, and “sexed quotation marks,” returns 6 hits. Does anyone know the source of the term? I know I didn’t imagine it. It must have been a teacher or editor somewhere along the line who used that phrase, and it stuck with me. But it may just be that I am a typographical pervert.