Exactly: and the key point that always bears discussion in pre-game is that it matters only where the K recovers the kick, not how it got there. As AB says, if K recovers the kick behind the neutral zone, the ball's status is as if they had just snapped it.
Well, it kind of does matter HOW it got there.
EXAMPLE: 4th and 10. K punts, ball crosses the NZ, rebounds behind the LOS, is picked up by K, who runs for 5 yards. Legal play, but ball will go over to R for K failing to reach the LTG.
Compare that to: 4th and 10. K punts, ball crosses the NZ where it bounces off an R player, rebounds behind the LOS, is picked up by K, who runs for 5 yards. K will now get a new series, 1st and 10. Had K thrown an incomplete pass after recovering the ball, they STILL would get a new series. But if they punt again, no new series unless K recovers after R once again touches the punt after it crosses the NZ.