Green’s Dictionary of Slang

1. the anus [SE round + -eye sfx].

1950		Goldin et al. DAUL 181/1: Roundeye. [...] 2. (P) The anus.
a.1963		‘The Fall’ in D. Wepman et al. Life (1976) 83: She had a good round-eye, and that’s no lie; / How the trickhouse door would swing!
1972		C. Bukowski Erections, Ejaculations etc. 255: God! That’s one UGLY-lookin’ roundeye if ever I saw one!
1979		Maledicta III:2 232: Still more words of this fucking vocabulary are [...] roundeye.

2. anal intercourse.

1970

(con. 1930) Gaddis & Long Panzram (2002) 241: Oh, how I love my roundeye!

3. a male homosexual [metonymy].

1950		Goldin et al. DAUL 181/1: Roundeye. 1. (P) General term for passive pederasts; any passive degenerate.
1969		‘Iceberg Slim’ Mama Black Widow 230: I was an experienced ‘round eye.’.

4. a white person, as opposed to an East Asian person [as opposed to slant n. (4)].

1967		J. Breslin World of Jimmy Breslin (1968) 136: People [...] only see this big round-eye standing over the six poor Viet Cong prisoners.
1968		Army Times ‘Army Talk in Vietnam’ 10 Apr. n.p.: roundeye: Caucasian woman.
1976		P. Conroy Great Santini (1977) 253: I’ve never killed a round eye in my whole career. I’ve majored strictly in slants.
1977		‘John le Carré’ Honourable Schoolboy 140: Very nice blonde party he had with him. Roundeye.
1990		(con. c.1970) G. Hasford Phantom Blooper 5: The Phantom Blooper was a round-eye, tall and white, with blond hair, wearing black pajamas and a red headband.
2001		(con. 1964–8) J. Ellroy Cold Six Thousand 314: O-heads boocoo. O-heads in orbit. Slants and some round-eyes. One jigaboo.