During play, pause the game and enter the following:

Restore health:
Up, Down, Up, Down, X, X, X.

Show score:
Right, Right, Right, Left.

One extra Kuji:
Left, Left, Left, Right, X

Full Kuji:
Hold L + R and press Left, Left, Left, Right, X.

100 extra points:
Hold White + Black and press Right, Right, Right, Left.

New move:
Hold R + Black and press Up, Up, Down, Down. Release R + Black and press X, X, L, R.

Stronger offense:
Hold R + White and press Up, Down, Up, Down. Release R + White and press X, X, X.

All characters:
Hold White + Black and press Up, Right, Left, Down. Release White + Black then press L, R.

Through The Portal extra mission:
White, Up, Black, Down, L, R, Left

B-side audio:
Hold L + R and press Down, X, X, Up, X, X, Left, X, X, Right, X, X. Now go into the Options menu and select Audio/Language to find the new B-side option.

Enter the following at the Mission select screen:

All story mode missions:
White, White, L, R, Right, X, click left stick, click right stick

All layouts:
Click right stick, click left stick, L, R, White, Black

Enter the following at the Item select screen:

All ninja items:
Hold L + R and press Up, Down, Up, Down, X, X, X, Left, Right, Left, Right, X, X, X.

Increased ninja items amount:
Hold L + R and press Up, Left, Down, Right, X, X, X.

Infinite capacity:
Hold L + R + White and press Up, Up, Down, Down, Left, Right, Left, Right, release White and press X, X, X.

Thanks to Revolution readers Jumanji, wallace gromit, farrah and Jake676!