A native of Dedham, Massachusetts, Twitchell attended Dean Academy before attending Rutgers. During his college years, he was an All-American defenseman, an honorable mention All-American football player, and the recipient fo the Donald Coursen Outstanding Athlete Award.
Upon graduation from Rutgers in 1935, Twitchell taught and coached at North Plainfield High School, Sewanhaka High School, and Hofstra University. Two of his Sewanhaka teams won Metropolitan Interscholastic Lacrosse Titles.
In 1947, Twitchell returned to Rutgers, and in 1950 he became head lacrosse coach in addition to his duties as varsity football line coach. He compiled a 12-year record of 90-39-1, and was awarded the Morris Touchstone Coach of the Year Award in 1958 and the United States Intercollegiate Lacrosse Association Man of the Year Award in 1962.
Twitchell has been president of the USILA and USLCA, Chairman of the North/South All-Star Game, Chairman of the USILA Long-Range Planning Committee, and he is an honorary life member of the USLCA. Twitchell served as director of athletics at Rutgers from 1961-1972. He was inducted into the US Lacrosse New Jersey Chapter Hall of Fame in 1997.