Does $L_1L_2 \notin RE$ imply $L_2L_1 \notin RE$?

Question

Given two languages $L_1, L_2$ such that $L_1L_2\notin RE$, is it always true that $L_2L_1 \notin RE$?

I wasn't able to prove it or find a valid counterexample.

Answer 1

Let $A \subseteq \mathbb{N}$ be an arbitrary subset containing $0$. Define $L_1 = \{0^n 1 : n \in A\}$ and $L_2 = \{0^n : n \in \mathbb{N}\}$. Then $A$ reduces to $L_1L_2$, but $L_2L_1 = \{0^n1 : n \in \mathbb{N}\}$.