Two tugboats on each end but opposite side of the barge and pushing it so as to rotate the barge; of course the tugboat operators will have to adjust for the strong current.
Actually, a serious answer is possible, assuming that the barge is alongside a dock.
Just tie a hawser between the downstream outboard corner of the barge and a bollard on the dock not far upstream or downstream.
It would be wise to also tie off the shoreside of the barge at the midpoint and the upstream shoreside corner with long hawsers that can be let out gradually to control the rotation.
Then push the upstream end out in the current and let the current do the rotation, using the thrusters to buck the current.
It gets more complicated if something is docked downstream less than a barge length away. Absent that complication, then you just pull the new upstream end, upstream to the original location.
If you're in the middle of the channel, well, if you WERE in the middle of the channel, you're screwed anyway, and maybe you can allide with something soft before you reach New Orleans....
My shiphandling is notoriously ropey, but one possibility could be to exploit irregularities in the current: In confined waters, there are often slack patches and if you stick your bows into one of those while the rest of the barge stays in the main flow, you are likely to swing a long way quite fast. When it takes you by surprise, the outcome is embarrassing, but as a deliberate move...
