SUMMATION

Let the numbers be part of a set $"S"$ .

Now, we see that, for every number $\overline{a_1a_2a_3a_4}$ of $S$ , the exists another member of $S$ , $\overline{b_1b_2b_3b_4}$ such that $\overline{a_1a_2a_3a_4} + \overline{b_1b_2b_3b_4} = 5555$ . For example, $1324$ can be paired with $4231$ , $2314$ can be paired with $3241$ .

Now, we see that there are $4! = 24$ numbers in $S$ . Since every pair contains 2 numbers, there are a total of 12 pairs.

Since the sum of each pair is 5555, the sum of all the numbers must be $12 \times 5555 = \boxed{66660}$ .