My girlfriend raised two good points yesterday regarding the placement of people at the Last Supper:
(1) Judas sitting on Jesus’ right wouldn’t mean that he was dipping with his left hand. He’d just be reaching across his body to dip. This was pretty obvious, once she said it. I don’t know how I missed that.
(2) Even sitting on Jesus’ left side would be a huge honor. In Mark 10:37, James and John ask to sit on either side of Jesus, but He tells them (in v. 40) that it is an honor not for Him to give.
It seems, then, that the most probable case is that John does end up at the right side of Jesus (which I imagine James was not thrilled about, since he seemingly didn’t get the left side). Judas probably sat on His left, which was a place of earthly glory, but foreshadowed eternal damnation. That’s a powerful illustration of Mark 10:31 (which perhaps significantly, immediately preceeded the left hand / right hand discussion), if true (and I should mention here that the Church has never claimed for sure that Judas is in hell, although Matthew 26:24 seems like a clue).