Calendar Cubes Puzzle

How many different ways are there to assign the ten digits of Arabic numerals (0 to 9) to each face of two cubes to ensure that we can arrange both cubes on any day such that the front faces of the cubes show the current day of the month?

For example, on February 9 the cubes would be placed side by side such that the front face of the cube on the left side shows 0 and that of the one on the right side shows 9.

Two ways of assigning the digits to the faces of the cubes are considered different if and only if it is not possible to get one assignment from the other by performing one or more of the following operrations:

1. Rotating (reorienting) the digits with respect to the faces they belong to.
2. Rotating the cubes.
3. Swapping the cubes.

See the comments page for the solution.