Cubbage was traded to the Minnesota Twins in 1976. On July 27, 1978, he hit for the cycle against the Toronto Blue Jays. Those that attended the game or watched on television called it an "accidental cycle". In the bottom of the 2nd inning, Cubbage hit a 1-0 pitch to right field, and was thrown out at third trying to leg out a triple; he was therefore credited for a double on the play. He subsequently hit a home run (4th inning), single (5th inning), and triple (7th inning) to complete the cycle.