The punt-out, also called a kick-out, was after a touchdown. Either you tried the point after from the point opposite where the ball went into the end zone (remember the goal posts were on the goal line), I think two yards out (three in 1927) was where it was spotted, and it had to be place kicked (drop kicking the blimp ball was easier). Or the scoring team could punt-out. A free kick to better center the ball. But if mishandled the other team could recover and start play from there. Rutgers botched one and lost a big game (somewhere around 1917) and so many bellyached that that rule was changed in 1919, for 1920 season (NCAA). Walter Camp in his syndicated column said it made things fairer, but in a column in New York Times actually disliked the rule change, saying that centering the XP attempt would make it near automatic.
Some of the rules mentioned have not been changed and some that have, it's been pretty recently. The NFL changed the rule where one could drop kick past the line of scrimmage and even though no one had did it in decades they changed the rule a few years ago.
I actually looked through my kicking page. Some will surprise you:
Aha! • • The scored upon team has the option to kickoff or receive. This is rule in NFL and National Federation of High School rules (NFHS). The NCAA changed this rule in 2003, the scoring team being required to kick. The last time this rule was used to kick off after being scored upon is undetermined so far.. When Georgia Tech beat Cumberland 212-0, it was used a lot. Cumberland figured kicking off gave the Rambling Wreck worse field position than when Cumberland fumbled or punted after they received a kickoff and went 4 and out.
So it could still happen in the NFL!! Unless someone knows better. Why bother changing it when no one would do it?
In Canada and in the CFL, they still can punt a punt back while play continues.
Here is a video of an Canadian Football League example. There are 4 kicks on this play. It is the last play of a tie game, and relates to how Canadian kicking teams can score a point if the kicked ball is not brought out of the end zone by run or kick:
http://youtu.be/d5BFaykcxGg
The NFHSA allowed field goals on kick-offs until 1948. It may still be a rule in Philadelphia PAL League/
On 9/25/2015, Luis Aranda of Robert E. Lee school of Midland Texas kicked an extra point that bounced off a referee's head and went over the crossbar and through the uprights, for the point. See it here:
http://video.businessinsider.com/5819bd ... c3b2e.webm
Until 1932, on a kick-off or free-kick, the receiving could send 10 players running back to where the kick was caught and form a flying wedge on the return, a “thundering herd”.
On 11/11/1911, Princeton beat Dartmouth when two drop kicks, by DeWitt, bounced over the crossbars and were called good for 3 points. The referee said there was no rule against it. In 1912 there was such a rule
An additional offensive kick was the “quarterback-kick”, or onside-kick from scrimmage. 1st used by Minnesota in 1892; it was the forerunner of the forward pass. A form of quick kick, it was usually kicked just after the center snap as the quarterback dropped back. The QB or any back behind him could recover a kicked ball. Coach George Woodruff of Penn also came up with the QB-kick in 1892. The quarter-back would kick the ball and his backs would try to catch it in the air or pounce on it after it hit the ground. The first quarterback kick by Penn was by Bucky Vail against Lafayette 11/16/1892. Also called an offside kick at times, it was not totally eliminated until 1922.
On punting back a punt I found only one example:
Oct. 22, 1933- Harry Newman of New York Giants punted to Shipwreck Kelly of the Brooklyn Dodgers who punted it back to the Giants who lost a yard in the exchange.
When the goal posts were on the goal line, a punt out of the end zone that hit the crossbar or goal posts and bounced back and out of end zone was a safety.
In Canada, it is illegal to kickoff with a drop kick