A symmetry property of some harmonic algebraic curves

The aim of this note is to give a surprising symmetry property of some harmonic algebraic curves: when all the roots $z_i$ of a complex polynomial $P$ lie on the unit circle $\text{U}$, the points of $\text{U}$ different from the $z_i$, and such that $\text{Arg}(P(z))=\theta$, form a regular $n$-gon, where $n$ is the degree of $P$.