Unfortunately, Aristophanes doesn't give any particular name for these pairs: he just calls them "people" or "humans" (ἀνθρώποι). He calls the half-man-half-woman one "androgynous" (ἀνδρόγυνος), but that's a common enough word nowadays that it wouldn't be a clear reference.
But later, he gives the origin of each gender:
ἦν δὲ διὰ ταῦτα τρία τὰ γένη καὶ τοιαῦτα, ὅτι τὸ μὲν ἄρρεν ἦν τοῦ ἡλίου τὴν ἀρχὴν ἔκγονον, τὸ δὲ θῆλυ τῆς γῆς, τὸ δὲ ἀμφοτέρων μετέχον τῆς σελήνης, ὅτι καὶ ἡ σελήνη ἀμφοτέρων μετέχει· περιφερῆ δὲ δὴ ἦν καὶ αὐτὰ καὶ ἡ πορεία αὐτῶν διὰ τὸ τοῖς γονεῦσιν ὅμοια εἶναι.
And this is why there were three genders, like I described: because males were, in the beginning, born from the sun, and females from the earth, and androgynes from the moon, because the moon is androgynous too. They were round, and moved in circles, just like their parents.
(Mythologically, Helios is male, Gaea is female, and Selene is both male and female according to some sources.)
This is probably the easiest part to reference: gay men descend from the sun/Helios, gay women from the earth/Gaea, and straight people from the moon/Selene. And originally they looked like little planets because of this parentage.
He also calls them "complete" (whole, entire) at one point:
Τοῦτο γάρ ἐστι τὸ αἴτιον, ὅτι ἡ ἀρχαία φύσις ἡμῶν ἦν αὕτη καὶ ἦμεν ὅλοι· τοῦ ὅλου οὖν τῇ ἐπιθυμίᾳ καὶ διώξει ἔρως ὄνομα.
For this is the cause of it all: that our original nature was like that, and we were complete, and now the desire and the pursuit of that completeness is called "love".
But this is the closest he comes to giving a name to the original form of humanity.