Green’s Dictionary of Slang

(Irish) a man with one testicle.

[1900–10		Stephens & O’Brien Materials for a Dict. of Aus. Sl. [unpub. ms.] 130: RIG: a proud-out or faulty cut stallion, perhaps left with one testicle].
1994		(con. 1930s) K.C. Kearns Dublin Tenement Life 152: Now her husband, a fine cut of a fella, it was commonly known to the menfolk around there that he was a ‘rig’, a man with one ball.