Rivers, the 22-year-old son of coach Doc Rivers, averaged 7.1 points, 2.0 rebounds and 1.7 assists in 41 regular-season games after arriving from Boston in a January trade. He became the first player in NBA history to be coached by his father.
Rivers signed a reported $6.4 million, two-year deal that includes a player option in the second year.
He appeared in all of the team's 14 postseason games, averaging 8.4 points and 1.1 assists.
Aldrich joins the Clippers as a backup to DeAndre Jordan, who re-signed with the team after reneging on a verbal agreement to join Dallas.
Aldrich averaged 5.5 points and 5.5 rebounds in 61 games last season for the Knicks, where he played for two years. The 26-year-old has played with Oklahoma City, Houston and Sacramento.