1 - A handjob, delivered through the boxers, usually in a situation in which the giver of the birdcage believes the recipient's genitals to be dirty or otherwise inadvisable to touch. A Chi-Phi-ism. Can be used as a verb.
2 - Metaphorically, anything unsatisfying or upsetting.
3 - A stock phrase, "Worse than a birdcage," used to connote extreme irritation.
1 - Man, this broad gave me the birdcage right on the dance floor. Sir!
2 - Those fucking cops were giving me the birdcage.
3 - The party was worse than a birdcage, bro, everyone was completely state.