In this article we complete the classification of infinite affine Coxeter group types with the property that every double coset relative to the first parabolic subgroup is represented by an involution. This involutive double coset property was established earlier for the Coxeter groups of type C~2 and G~2, we complete the classification by showing it also holds for type F~4 and the types C~n for all n. As this property is inherited by all string C-groups of these types, it follows that the corresponding abstract regular polytopes will have polyhedral realization cones.