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

On symmetric Willmore surfaces in spheres II: the orientation reversing case