wrens used to feature in some Christmas, and probably pagan, practices - I think some people used to try to kill one on the day. And geese were what you ate before turkeys came into vogue.
I'm not certain if that actually amounts to ' symbolising' Christmas, though. But I don't know of any other yule bird.
Incidentally, robins apparently came to do the job because the red breasts reminded people of the red coats postmen used to wear, so in Victorian times they were shown as postmen delivering Christmas cards.