  In this paper we provide a systematic treatment of Willmore surfaces with
orientation reversing symmetries and illustrate the theory by (old and new)
examples. We apply our theory to isotropic Willmore two-spheres in $S^4$ and
derive a necessary condition for such ( possibly branched) isotropic surfaces
to descend to (possibly branched) maps from $\mathbb{R} P^2$ to $S^4$. The
Veronese sphere and several other examples of non-branched Willmore immersions
from $\mathbb{R} P^2$ to $S^4$ are derived as an illustration of the general
theory. The Willmore immersions of $\mathbb{R} P^2$, just mentioned and
different from the Veronese sphere, are new to the authors' best knowledge.
