The emission of neutron pairs from the neutron-rich N=12 isotones C18 and O20 has been studied by high-energy nucleon knockout from N19 and O21 secondary beams, populating unbound states of the two isotones up to 15 MeV above their two-neutron emission thresholds. The analysis of triple fragment-n-n correlations shows that the decay N19(-1p)C18∗→C16+n+n is clearly dominated by direct pair emission. The two-neutron correlation strength, the largest ever observed, suggests the predominance of a C14 core surrounded by four valence neutrons arranged in strongly correlated pairs. On the other hand, a significant competition of a sequential branch is found in the decay O21(-1n)O20∗→O18+n+n, attributed to its formation through the knockout of a deeply bound neutron that breaks the O16 core and reduces the number of pairs.