Both teams announced the trade Friday night.
The addition of Neal signals the Predators' emphasis on boosting their offense under new coach Peter Laviolette as they attempt to return to the playoffs after a two-year absence. The Predators have had a reputation as a defense-oriented team for virtually the franchise's entire history.
The 26-year-old Neal had 27 goals and 34 assists in 59 games for the Penguins this season, which marked the sixth straight year he scored at least 20 goals.
Neal scored 40 goals in 2011-12 and had 21 goals in only 40 games during the 2012-13 season. He has 161 career goals and 154 assists in 413 career regular-season games, and has been a force on power plays.
Neal had been with the Penguins since 2010 after beginning his career with the Dallas Stars.
Hornqvist, 27, had 22 goals and 31 assists in 76 games this season, which marked the fourth time in his career he reached the 20-goal mark. He has 106 goals and 110 assists in 363 career regular-season games, all with Nashville.
Hornqvist's best season came in 2009-10, when he scored a career-high 30 goals.
''In this deal, we got to drop our cap a little bit with Hornqvist,'' Penguins general manager Jim Rutherford said. ''We like Hornqvist as an all-around player. Hornqvist plays at the net, goes to the net, works the corners. We like his all-around play.''
Spaling, 25, had 13 goals and 19 assists in 71 games for Nashville this season. He has 40 goals and 44 assists in 207 career regular-season games.
''We like Spalding, he can play all three positions on the third line,'' Rutherford said. ''I think we were just trying to change the mix of our team a little bit and get a little different type player.''