Not really a Gradius game, but it does draw on Gradius and Irem's R-Type and X Multiply. Awesome shooter from the '90s.
Xexex has had differences in its releases. In the Japanese version, 2 players take turns playing, and uses an extra lives system. International versions allow for cooperative play and uses a health bar instead of lives. Name entry in Japanese versions is where you put in your initals, gender, and Chinese zodiac at a shrine, where overseas you just put in your initials below your score while the continue counts down.
The version played here is the European version; the game was released as Orius in North America. I like sticking with the original name sometimes.