It's not a left-handed mouse (trackball); the buttons are operated with your thumb and the ball with your index (and middle) finger
Turns out on that model (Elecom huge if that is indeed what it is) you actually are operating the track ball with your index finger. In your defense I thought the exact same thing when I saw the scroll wheel on the left side.
Nice mouse, I have the same :)
First thing I noticed. What model?
Search for Elecom Huge