The deal was announced Friday.
Richardson averaged 10.2 points and 4.4 rebounds in 72 games with the New York Knicks last season. He's returning to the Clippers, where he played from 2000-04 after being drafted by the team out of DePaul.
The 29-year-old guard-forward was acquired by Memphis on June 25 in exchange for Darko Milicic. Richardson originally left Los Angeles as a free agent and went to Phoenix, where he played one season before joining the Knicks.
Randolph was acquired in a trade with the Knicks in November 2008. The forward averaged 20.8 points and 10.1 rebounds in 50 games for the Clippers last season.
But he had a rocky season off the court, including his father's death in March.
Randolph was suspended for two games by the team after being arrested for investigation of drunken driving in April. He was suspended for two games for punching a Phoenix player in February.
In 2003 while with the Portland Trail Blazers, Randolph was arrested after an officer noticed the smell of marijuana coming from his vehicle.
