The 76ers sent Speights to Memphis, the Grizzlies moved Henry to New Orleans and the Grizzlies and Hornets each sent a second-round draft pick to Philadelphia.
The six-foot-six Henry, in his second NBA season out of Kansas, averaged 4.3 points and 1.0 rebound in 38 games with Memphis.
"We are delighted to welcome Xavier to our team and the New Orleans community," Hornets general manager Dell Demps said. "We hope Xavier will develop into a good basketball player that will contribute to the success of the New Orleans Hornets this season and into the future."
Henry became the first player in Kansas history to leave for the NBA after his freshman season.
The six-foot-10 Speights is in his fourth NBA season out of Florida and has averaged 7.2 points and 3.7 rebounds. The Grizzlies hope he will help make up for the loss of Zach Randolph, the star forward expected to miss six-to-eight weeks after tearing a ligament in his right knee.
Philadelphia general manager Rod Thorn said trading "will give us additional assets moving forward, including a trade exception, which will help us as we look to continue to improve our team."
Teams with a trade exception have up to a year in which they can acquire more salary in other trades than they send away.