Even gaps between six pairs of coloured blocks

Hannah owns $12$ blocks, two of each of $6$ colours, and lays them in a row in a uniformly random order. What is the probability that, for every colour, the number of blocks lying strictly between the two blocks of that colour is even?

$\tfrac{16}{231}$. An even gap means the pair occupies one odd and one even position. So each colour gets exactly one of the six odd slots and one of the six even slots: $6!\cdot6!$ arrangements out of $\tfrac{12!}{2^6}$, giving $\tfrac{16}{231}$.