ISL 2015 A5

Let $2 \mathbb{Z}+1$ denote the set of odd integers. Find all functions $f: \mathbb{Z} \rightarrow 2 \mathbb{Z}+1$ satisfying $$f(x+f(x)+y)+f(x-f(x)-y)=f(x+y)+f(x-y)$$ for every $x, y \in \mathbb{Z}$.