I know some people don't take the twine off. I do for all the reasons listed above. I've seen cows ingest it. (that can't be a good thing) I've had it wrap around a foot, and cut off circulation, and then you have to catch the animal and get it cut off the leg. I've had it wrap around finnal drives, drop axels, pickup drive shafts, etc. and it will knock out a seal every time. Oh did I mention the amount of time it takes to get a large quantity of twine unwrapped from the back end of a manure spreder?
I'm sort of extream the other way. It takes me a little longer to feed cows, but I won't even cut the twine.
I unwrap the twine off of the bailes the opposite way it went on. That way I'm more certian to get all the twine. It seems like every time someone comes out and helps me feed, they use their pocket knive and slice down the baile and pull the twine off. I can almost certianly go around behind them and find more twine they missed by not grabbing a little bit here or there. Plus the extra little bit of time it takes me to unwrap the bails that way, gives me a little bit more up close and personal, face to face time with the girls, seeing if anyone has pink eye, a limp, a lump, etc.