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 (rotation can be used instead in Tetris Best because it has kicks), or an O-spin (single only --in Tetris Best O-spin doubles are possible because they rotate and kick). Note that Tetris Best 2 has floorkicks only, even though it's supposed to have mild wallkicks.