Diagonal movement exists in Tetris & Dr. Mario, Tetris Best and Tetris Best 2. The piece can move downleft or downright.

In Tetris & Dr. Mario, you must use soft drop and left/right to move diagonal.

In Tetris Best and Tetris Best 2, this can simply be done by pressing left or right to a tight space. It counts as a spin.

It allows O-spins to be done, if you count this as a spin. In Tetris Best 2 this is the only way to do Z- or S-spin doubles (in Tetris Best this could also be done with a wall kick), or an O-spin (single!).