i got another:
0 1 2 3 7 8 on first
0 1 2 4 5 6 on second

1,2,3,4,5,6 (for ones place)
1,2,3,7,8,9 (for tens-place - there is no date above 31, therefore no 4,5,6,etc. is needed)

Sorry - forgot about 0s

I came up with the same solution as the software engineer

012345 on the first one
012678 on the second
you dont need a 9 cos the 6 can be turned upside down

Actually the problem can be solved like this:
the cubes have to have 0,1,2 each
1st cube: 0,1,2,_,_,_
2nd cube: 0,1,2,_,_,_
and you can fill in with 3,4,5,6,7,8 in the remaining blanks however you want (ofc 6 can be turned), so you actually have 20 different possible solutions (considering that 0,1,2,3,4,5 is the same with 0,1,2,4,5,3) and we dont add the fact that some ppl might give the same solutions with a 9, which would be another 20 solutions

thanks i really needed these solutions of a paper i am writhing

first 0,1,2,3,4,5
second 0,1,6,7,8,9

Fangola: Your answer is incorrect as you cannot make 22 with the numbers given. You don't need 9 because 6 can be turned upside down.

I think the 6 being turned upside-down is a dodgy answer to a logic puzzle.