It stems from the lay out of the parties in the Jacobin chamber

It's that way in the US Congress as well. The Dems sit on the left (facing the speaker) and the Reps on the right. (I wonder where Jim Jeffords sits.)

Does anyone know when the US developed this custom? I wouldn't think that the early Congress would have copied what the Brits were doing, seeing as the new nation was trying to be completely different.