Before it evolved into interleague play, the Mets and Yankees engaged in the Mayor's Trophy Game. It was a series of exhibitions for charity. The games did not count in the standings.
It started in 1963 at Yankee Stadium and had a bit of controversy as Mets manager Casey Stengel, in what was viewed at a slap at his former team, elected to go with his best pitcher, Carl Willey. The Mets won, 6-2. In later years, both teams would use the game to try out minor league pitchers.
The Mayor's Trophy series went from 1963-79 and 1982-83. Veteran players hated the game. The most infamous game took place in 1978 when Yankees third baseman Graig Nettles threw the ball way over the head of first baseman Chris Chambliss in the 11th inning. Some viewed that as an attempt to finally end the game (which the Yanks won in 13 innings). Yankee pitcher Sparky Lyle said as much in his book, The Bronx Zoo. Nettles denied it.
The series ended with the Yankees holding a 10-8-1 advantage.