Ok, ladies and gentlemen...I'm not responsible for this one, but curious nonetheless...
End of 1st period, technical called on red team as teams go to huddles. So we should have two shots for black and a throw-in for black at the division line, right? Only the refs in this situation give black their 2 shots, but for some strange reason go to the possession arrow for the throw-in (which is pointing to red!) So they give the ball to red (EEK!) and play resumes and the arrow is switched to black. Obviously the ball should have been given to black for the division line throw-in to start the second period. However, I'm not sure about the arrow. If things were done right and black got the throw-in (with the arrow pointing to red) do you switch the arrow following the throw-in? Thanks for all your input :)

Black's throw-in to start the second quarter is part of the penalty for Red's technical foul. The throw-in is not an Alternating Possession throw-in. Red retains the AP arrow for the next held ball or jump ball situation.

IT's not (entirely) clear from your post when the T was given. Either way the officials messed up.

If the T was *after* the first quarter ended, then the T was part of the second quarter and Black shoul dhave the throw-in as part of the T. That's what you addresses.

But, it's possible the T happened as the last thing *IN* the first quarter. I've known more that one official who thinks the throws for the T are shot after the break, so maybe that's what the officials did (the officials messed up by not shooting the throws for the T before the break). In this case, they got the throw-in correct, becuase the throw-in part of the penalty for the T doesn't carry over.