Is "being harmonic conjugate" a symmetric relation? The question is:
Prove or disprove the following: If $u,v:\mathbb{R}^2 \to \mathbb{R}$
are functions and $v$ is a harmonic
conjugate of $u$, then $u$ is a harmonic conjugate of $v$ (in other words, show whether or
not the relation of being a harmonic conjugate is symmetric)
I'm pretty sure i'm correct in saying it isn't a symmetric relationship... But I'm wondering if someone can think of a direct counter-example to prove me right. Or is there an algebraic way to prove this that's better?
Any help is appreciated, thanks!

Here's the simplest example I can think of. $f(z)=z$ is clearly holomorphic, but $g(z)=Im(z)+iRe(z)$ is not, because when you look at the second Cauchy-Riemann equation you get $1 \neq -1$.
