If you want to make Xbox 360 accessories you have to get approval from Microsoft first. Those are the rules. Unfortunately, the makers of the X-Arcade stick don't have Microsoft's blessing. However, they found a way to make their arcade stick work with an Xbox 360 and it's a creative solution.
First, you need to plug a PlayStation 2 adapter to your X-Arcade stick. Then you plug that adapter into an Xbox 360 to PlayStation 2 adapter and link a wired Xbox 360 gamepad to it. Finally, plug the daisy chained device into your Xbox 360 and like magic the X-Arcade stick and any other PlayStation 2 accessories become Xbox 360 compatible. Want to connect two players to a dual stick? Better get two adapters and two wired controllers.