'Catalysis' actually is an etymological opposite from 'analysis' with the '-lysis' ending coming from Greek lyein, meaning to loosen, a derivation of the PIE root leu-, meaning to loosen, divide or cut apart. However, in the given example, the 'ana' prefix does not mean up, nor does 'cata' mean down. 'Cata', in this case means throughout, corresponding to the original meaning of this word: dissolution (armies, governments etc.). It was then taken to mean the 'change caused by an agent that itself remains unchanged' by the chemist, Berzelius, then morphing it into meaning the quickening of a process. The meaning of 'ana' is very diverse in Greek, ranging from up to back to throughout, thus partly synonymous with 'cata'. Thinkers such as Aristotle used the Greek word analyein to mean 'solve a problem' by analysis. literally breaking up the problem, alluding to methodology.
After hours of tough research, I have found two words that are literal opposites: catadromous and anadromous, meaning going from river to sea to breed and vice versa, respectively.