USAJMO 2019 Problem 2

For which pairs of integers $(a, b)$ do there exist functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ and $g: \mathbb{Z} \rightarrow \mathbb{Z}$ obeying $$f(g(x))=x+a \text { and } g(f(x))=x+b$$ for all integers $x$ ?