The Perplexity of Haruhi Suzumiya is the first PlayStation 2 game released for the franchise. The gameplay is a mix of things varying from visual novel (game CGs accompanied by dialogue or text. See some CGs from the game), to a simple fantasy RPG, to a “princess maker“-type game. If you’ve got an affinity for the Japanese language, then head over to the official site. Here’s some stuff I grabbed from the site and digging around with google.jp
There currently isn’t any plans on porting this game …