Ball Hits Opponent's Backboard ...
 

Advantage ...
 

Just for fun: Fake a defensive opponent out of position?

Dribble ...
 

So it's a dribble?

4-4-4: A ball which touches ... an official is the same as the ball touching the floor at that individual’s location.

But the ball didn't bounce directly back to A1. It touched something else, the ring. The whole point of OP was if the ball hit the ring instead of the backboard only, would it be legal to start a dribble? Some seem to indicate "legal" since the rule specifically says backboard only. I am asking if the ball hits the backboard, then the ring, then goes to A1 would it be legal for A1 to start a dribble since the rule specifically says the ball (from the backboard) must return directly to A1 which it did not.

Yet Another Rabbit Hole ...
 

My opinion: The pass against B’s backboard was still the start of a dribble, use "normal" rules to decide what can, or cannot, legally happen next.

Keeping only the ring (not ring and backboard) in the situation makes the interpretation less complex, not a lot, but a little.

http://www.dictionary.com/e/wp-conte...abbit-hole.jpg

This one is easy. Backcourt violation in both cases (a) and (b).
Why? We have a rule which tells us that the opponent’s basket is part of a team’s backcourt. 4-13-2 “…A team’s backcourt consists of the rest of the court, including the entire division line and the opponent’s basket and inbounds part of the - opponent’s backboard.“

As you have written, case (a) is legal due to a casebook ruling. I’ll note that the rationale of that casebook play was modified just this year to make tossing the ball off one’s own backboard a try for goal.
Case (b) is more difficult because the ball striking the ring is not automatically considered a try for goal. For example, a pass which strikes the ring would not reset the shot clock should one be in use. Therefore, if the official does not deem this action a try, then it must be treated as a dribble which does not strike the floor. Now we must consider if the player who threw the ball lifted his pivot foot AND returned it to the floor prior to touching the ball again. If so, then an illegal dribble violation has occurred (what the NFHS casebook now labels a traveling violation, but used to list as an illegal dribble). If not, then the action is legal.

Why isn't it just "tossing the ball in the air and catching it" -- as in the first play I posted?

Clarification ...
 

... while keeping pivot foot in contact with the floor?

But it's more than that. If it hits the opponent's ring, it appears to be a dribble (or a part of a dribble).

A Team’s Backcourt ...
 

Nice citation Nevadaref.

and, part of why I'm asking -- if the basket is part of the backcourt, then shouldn't the ball hitting the basket be the same as the ball hitting the backboard?

Transitive Property ...
 

Then a ball hitting the opponent's ring, flange, backboard, or frontcourt floor would all be treated the same, as a dribble (or part of a dribble).

Nice rationale bob jenkins.

Do we also include the opponent's net? I have seen cross court passes in one's backcourt hit the opponent's net.

The ball is unlikely to rebound to the player after just hitting the net -- but I agree the rationale should be the same. Maybe the NFHS will change this, or maybe someone can submit a change

Temporarily Caught In The Net ...
 

While the ball may not rebound, it could get temporarily caught in the net and just fall straight down (maybe with a little pendulum back track action) to the floor. If the passer gets to the ball first, picks the ball up, and starts a dribble, it's an illegal (double) dribble.

Call that and a head coach might end up sitting on a cold bus in the parking lot.

In Mike Goodwin's games the coach could get severe frostbite.

https://tse2.mm.bing.net/th?id=OIP.P...=0&w=226&h=170

It could happen, probably won't, but I would not bet my house that it could never happen.

If we're going to go down rabbit holes, we might as well go all the way down the rabbit holes.

Yes, that would be fine too. The two thoughts are equivalent in this instance.

Since this action is not tossing the ball to another player it cannot be a pass and as the player is not throwing for goal (certainly not at the wrong hoop), it cannot be a try. That leaves us with the potential start of a dribble. One which has yet to strike the floor. So the player may not move his pivot foot to another location and then touch the ball again prior to it contacting the floor (or the backboard or an official).

I agree with Bob that the ball striking the ring at either basket when clearly not a try for goal must be treated in exactly the same manner.