How much actual possession and control is needed for a pass by a defender to an attacker in an offside position to negate offside?

The following scenario happened to me recently in a high level mens amateur game where I was AR2. Right half sends a long ball intended for his teammate in an offside position behind the defensive teams back four. Pass however is woefully short and left fullback jumps to head the ball. Inexplicably, instead of heading the ball upfield, he executes a twisting header directing the ball back and square to where he believes his teammate/sweeper will be. Instead, it goes directly to the player who was in an offside position at the time the ball was played forward. I keep my flag down thinking it was a pass back. I’m told later by senior referees that it was a “continuation” of the original play and my flag should have went up. Apparently, a twisting header is NOT enough actual possession and control to be considered a pass back. Is this correct? How about if the defender was actually heading the ball downwards and back towards the offside attacker and/or where he though the sweeper may be? The impression I got from the National Level ref working the middle is that the defender basically had to have controlled the ball with his feet and passed it back.

USSF answer (October 12, 2010):
In all such cases, only the referee can make that decision, in this case with input from other officials on the game. If the defender was able to exercise as much control as you suggest, then there should be no doubt in your mind — or anyone else’s — that he had both control and possession. In such a case, there is no need for a flag, as the attacking player would appear to have had no influence whatsoever on the play.

Two further comments:
• We hope you misunderstood the statement by the “National Level ref,” as it is wrong. “Control” (for purposes of analyzing an offside position scenario like this) is NOT defined by “had to have controlled the ball with his feet and passed it back.”
• If the defender had controlled the ball with his feet and then kicked it to the goalkeeper, we get into another infringement of the Laws, provided the goalkeeper played the ball with his hands.