Entropy of Sum vs Difference of Random Variable

Question

I am looking for a proof of the following fact

Let X and X' be i.i.d on {0,1,2}(not necessarily uniform). Prove that

$H(X + X' mod\;3) \leq H(X - X' mod\;3)$

where $H()$ is the standard Shannon entropy.

A short(or long) proof or any pointers would be very helpful

Thanks in advance