This picture is also in Georges Bordonove's book, Les Rois qui Ont Fait la France: Henri III, but shown the other way i.e. Henri III is pointing to the left. 

Source of picture in Bordonove book is a miniature from Musee de Condé, Chantilly. My image source: Lumiere

The next portrait, from ca. 1850, shows him in a frivolous setting with monkeys and parrots - fashionable animals of the times, but showing his non serious side