Main Menu Cheats:

All of these hacks start with the sequence Up, Down, Left, Right, R1 followed by these
subsequences:
UP, DOWN - enable hardcore-mode (advanced physics)
DOWN, UP - enable hiding replay overlay (when this cheat is active, pressing the Circle
button in replay mode will hide the replay overlay)
LEFT, RIGHT - disable strict mode in beat-the-heat (allow hit walls)

Race Day and Single Race Cheats:

At the Race Day screen in Single Race or Head-to-Head, the following cheats are present Up, Down, Left, Right, R1, Up, Up: start a race with hornball enabled (press Up on D-Pad to fire a ball).

Up, Down, Left, Right, R1, Down, Down: start a practice session with hornball enabled.

Show Clean Replays:

At the main menu, press UP, DOWN, LEFT, RIGHT, R1, DOWN UP. After you enter the code, whenever you see a replay, you can press CIRCLE to view the replay with a clean screen.

Throw Tires:

After you've chosen your driver and track in Single Race, you'll be taken to the Race Day screen. Press UP, DOWN,LEFT, RIGHT, R1, UP, UP and you'll be taken to the race. Throw tires by pressing UP once you begin racing.

Richard Petty:

Complete all Heat challenges with a bronze or higher rating.

Hardcore realism mode:

Get a 100 point rating on a track to get the hardcore realism mode option for that track.