I've found a little trick, notable on the 8-bit side, that could also work on a SNES game if someone wanted to try it. It's like a pin alignment with the connector on the circuit board on the Twin to your game.
What I've found works (my prime example Tetris, I thought that my first cartridge was broken so I asked for a replacement, and the same thing happened with a different game - the game would load, it just came up in a garbled mess), is to insert the game from the right to left.
I've made an MSPaint instruction picture. I don't think I should have to type this all out, since seeing is a little better than just reading. The three lines represent the slot of your Twin - I made that game myself.

Hopefully that made sense. It's a good method to make sure your game and Twin are actually connecting correctly - I'm pretty sure most of you savy tech people probably knew all this, but this could be an idea for newbs to try if they have a problem with garbled screens.
EDIT: It looks like Photobucket resized the picture. I wonder if you can even read it, with your resolutions and such.
I'll just write the instructions.
1 - Put your cartridge in on the right side as much as it can, make sure the game is firmly in with the connector on the Twin circuit board.
2 - Push down on right side, connecting the games right side to the Twin's circuit board connector.
3 - You can either just slam the left side in, or gently let your game follow the connector by guiding it in. Make sure the left side is in now.
4 - Your game should be aligned with the connector in the circuit board. Turn on and see.
This should be a last resort. Always clean your cartridges!
