For more than a decade, doubt towards vaccines has become widespread in numerous countries, especially in France, raising major concerns for public health. Polarization of opinions on this matter, especially through social media, has been repeatedly observed, but details about the balance of forces are left unclear. In this paper, we study 107.923 French-speaking Twitter accounts. Two major asymmetries appear. Rather than opposing themselves on each vaccine-related controversy, pro- and anti-vaccine accounts focus on different vaccines and vaccine-related topics. Pro-vaccine accounts focus on hopes for new groundbreaking vaccines and on ongoing outbreaks of vaccine-preventable illnesses. Vaccine critics concentrate their posts on a limited number of controversial vaccines and adjuvants. Furthermore, vaccine-critical accounts display greater craft and energy, using a wider variety of sources, and a more coordinated set of hashtags. This double asymmetry can have serious consequences. Despite the presence of a large number of pro-vaccine accounts, fears raised by efficiently organized and very active vaccine-critical activists are left unanswered.