A perplexing and pointless cliche.
Rats that leave a sinking ship, spilled beans and drunk lords are at least clear and comprehensible images. Yet laughing on the other side of ones face has managed to achieve the exalted position of cliche without managing to mean anything at all. Even if one were able to switch one's laugh from one side to the other, one would still be laughing. Indeed, one would have the last laugh.
There are important and unexamined aspects of laughter. Is it, for example, possible to laugh like a drain up your sleeve? How do drains laugh in the first place? And what at?
Neither the OED nor Chambers nor Brewer's offer an explanation. Google has a strange and unverified assertion that it's to do with Janus, which doesn't work at all as Janus had two faces and might have laughed equally with both.
However, the OED did at least tell me that laughee means the object of laughter and that laughsome has the delightful meaning of "addicted to laughing". Perhaps, this was the affliction of the Laughing Murderer of Verdun whom I found mentioned in Brewer.
Picture snaffled from a fascinating article on Wittgenstein