A1 shoots towards A's basket. The ball hits nothing and is on the downward flight when A1 rebounds and sticks back in for 2. Legal or traveling?

Absolutely legal. No team control or player control on a try for a field goal. The ball belongs to no one, and anyone, including A-1, can recover and shoot.

Cha-ching!!

what about rebound air ball and dribble then shoot??? :)

a shot is a shot. makes no difference it it travels 2 feet or bangs of the boards- No team has pocession after a shot, so the rebounder is free to do what anyone is allowed once pocession is gained

Stew in VA
CVBOA

Why the confusion
 

As has been said when this play has come up in past threads, the reason that we have so much misunderstanding on this play is that the rule is exactly the opposite in the NBA. Catching your own airball is considered a traveling violation in the NBA. Now let's see what level of play is on TV the most? NBA. What level of play does the average fan/coach see most frequently? Probably NBA on TV. Hence, all the grief we take at the HS/College level for getting the play right.

The key is that the referee deem it a "try," per 4-40-2.

reguarding players/coaches/fans watching NBA.....
I had a situation right in front of Coach A (Jr High game), his player attempted a crossover dribble and in the process, caught the ball, ran 3 steps, switched hands and re-started his dribble. My whistle automatically blew because it knew something wasn't right, called the violation and we set up to go the other way. Coach A says, "that's legal, they do it in the NBA". I was in the process of handing B1 the basketball but pulled it back to clarify what he said. When he confirmed his conviction, and I got done laughing, B1 inbounded the ball and we went on.