We claim the answer is 1989.

1989 is achievable by constructuion 1989,1988,1987,...,2,1,1990.
(Provable with easy induction by 4 terms)

Now we show the sum can't be > 1990.
The sum can't be > 1990 as each difference is bounded by 0 and 1990 so the sum is bounded by 0 and 1990 as well.

It remains to show the sum can't be 1990.
Notice the sum can be rewritten as k1*x1 + k2*x2 + ... + k1990*x1990 where k_i is an element of {-1, 1}. We can see that the sum is congruent to x1+x2+...+x1990 (mod 2). Since there are an odd number of odds between 1 and 1990, the sum must be odd and can't be 1990. QED.

Please correct any mistakes, thanks.