Barring an administrative T, the direction of the alternating possession arrow is initially established with the game-starting jump ball. The possession arrow also shows which team will have the ball to start the 2nd, 3rd, and 4th quarter. The arrow does not change until the ball is inbounded.
The purpose of the alternating possession arrow is to show which team will receive the ball in case of a held ball, double foul, double lane violation on free throw attempt, a lodged ball (between rim and backboard), OOB when neither partner knows who touched it last. Another results if the official allows a team two free throws when it should have been a 1-and-1and the shooter misses the first shot and the lead official retreives the ball and realizes that it should only be a 1-and-1 situation. (NOT ME!

, this time...)
What are other situations? Any more?