This phrase always irked me, because when you're rotating something, it's not moving either left or right. The top of it moves right, the bottom moves left. Is it tightening or loosening?
Further, the only time a phrase might be needed to work out which way to tighten something is when it's unusually-positioned, for example upside down or facing away from you, and in those cases "righty tighty, lefty loosey" gives exactly the wrong answer (to the extent that it gives an answer at all, since "clockwise" and "right" are not the same thing).
I prefer the phrase "clockwise moves the thing you're turning away from you, anti-clockwise moves it towards you", which I'm hoping will catch on.