I hope maybe this picture will help explain what I am trying to figure out a little better.
See TOK's photo above. That is the proper orientation for a Betson Imperial trackball. The wires exit at the lower right when viewed from above.
The "drift" is normal, and it's due to the roller design. The Betson isn't alone, as pretty much all trackballs have this kind configuration internally. Some arcade games, like Marble Madness, actually change the orientation in software and position the trackballs in such a way as to maximize their efficacy based on the way the game is expected to be played.
If you position it incorrectly when you install it, there's probably a good chance you can swap some wires around and fix it, but not the way you showed it. The rollers with the encoder wheels must be positioned to react to motion in one of the 4 cardinal directions. Positioning as you have for the photo would probably require a special driver to translate it.
RandyT