I'm anti-adjuvant because adjuvants push your body's immune system into a Th2-heavy mode (antiparasitic/antifungal). Repeated adjuvant administrations are likely to contribute to immune disorders, especially Th2-heavy disorders and immunoproliferative disorders. Many, many autistic people have Th2-heavy (and generally imbalanced) immune systems. When exposed to immune stimulants, you'd expect these people to show additional symptoms. Adjuvants, by design, overstimulate immune systems. Repeated adjuvant exposure during infancy probably contributes to disorders of hyperactive immune systems.

Live virus vaccines are adjuvant-free. MMR vaccine, studied in the "vaccines cause autism" study and refuted by countless other studies, is actually adjuvant-free.

TL;DR -- Adjuvants are likely to overstimulate some peoples' immune systems, because they're designed to strongly stimulate peoples' immune systems.

Still better to use adjuvanted vaccines than be unvaccinated and get terrible illnesses though.