He pitched for eight different teams, including the San Diego Padres when they lost the 1984 World Series to the Detroit Tigers in five games.

Harris also was unusual in that, for periods during 1991–1993 while pitching for the Boston Red Sox, he achieved success while throwing curveballs almost exclusively.

A natural right-hander, by 1986 Harris could throw well enough left-handed that he felt he could pitch with either hand in a game, but his team would not allow this, keeping him from being a legitimate ambidextrous major league pitcher.

In the ninth inning, Harris retired Reggie Sanders pitching right-handed, then switched to his left hand for the next two hitters, Hal Morris and Ed Taubensee, who both batted lefty.

Harris's glove, which was custom built by Mizuno with an extra thumb so that it could be worn on either hand, is now on display at the National Baseball Hall of Fame.