Learn Finnish: Voit silittää jompaakumpaa kissaa.

The base form is jompikumpi (either). A unique feature of these dual pronouns is that both halves of the word change when you put them in a different case. Here, the word is in the partitive case, so jompi becomes jompaa and kumpi becomes kumpaa, combining into jompaakumpaa.

Why do we need the partitive case for jompaakumpaa kissaa?

The verb silittää takes a partitive object because petting is a continuous, ongoing action without a definitive end. Therefore, both the pronoun (jompaakumpaa) and the noun it modifies (kissaa) must be in the partitive case.

What would happen if I used kumpaakin instead of jompaakumpaa?

It would change the meaning from 'either' to 'both'. Jompikumpi implies a choice of one out of two (either this cat or that cat). If you said Voit silittää kumpaakin kissaa, it means 'You can pet both cats'.

I thought silittää meant 'to iron'. Is it the same word?

Yes, silittää means both 'to iron' and 'to pet' or 'to stroke'. The underlying meaning is to make something smooth. Context usually makes it completely clear which one you mean!