Another variant of PARTITION
Here is a reduction from PARTITION to this problem. Let $(a_1,\dots, a_n)$ be an instance of PARTITION. Assume that $a_1\leq a_2\leq \dots \leq a_n$.

Let $N$ be a “very large number”, e.g. $N = (\sum_{i=1}^n |a_i|) + 1$. Consider the instance 
$$\underbrace{N, \dots, N}_{5n \text{ times}}, N + a_1, \dots, N+a_n,\underbrace{4N, \dots, 4N}_{n \text{ times}}$$
of our problem. 


If there is a solution $x_1,\dots, x_n$ to PARTITION then 
$$\underbrace{1, \dots, 1}_{4n \text{ times}},-x_1,\dots,-x_n,x_1,\dots,x_n,\underbrace{-1,\dots,-1}_{n \text{ times}}$$ 
is a solution to our problem.
If there is a solution $(x_1,\dots,x_{5n},y_1,\dots, y_n, z_1,\dots,z_n)$ to the instance of our problem (which we reduced an instance of PARTITION to), then $\sum_{i=1}^n a_i y_i \equiv 0 \pmod N$. Thus $$\sum_{i=1}^n a_i y_i = 0. $$
That is, $(y_1,\dots, y_n)$ is a solution to PARTITION.