A solution that fits a generalized pattern:
[ 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 ]
x [ 1 3 5 7 9 11 11 13 13 15 15 17 17 19 19 21 23 25 27 29 ]
Reached by hand:
Began by looking at what would be equivalent puzzles for pairs of 2-faced, 4-faced, 6-faced and 8-faced dice, along with the assumption that the intervals between their faces' numbers would be symmetric because the resulting distribution is symmetric.
The intervals' symmetry could be either like ABBA and CDDC, as turned out to be the case, or like ABCD and DCBA, where the intervals of one die reverse the intervals of the other die.
2 faces (essentially a pair of coins with no alternative to the standard spot pattern): [1 2] x [1 2]
4 faces (easy): [1 2 2 3] x [1 3 3 5]
6 faces (the famous case): [1 2 2 3 3 4] x [1 3 4 5 6 8]
8 faces (not too tough, based on those above): [1 2 2 3 3 4 4 5] x [1 3 5 5 7 7 9 11]
20 faces was straightforward after 8 faces refined the pattern established by 2, 4 and 6 faces.