  The emission of neutron pairs from the neutron-rich $N\!=\!12$ isotones
$^{18}$C and $^{20}$O has been studied by high-energy nucleon knockout from
$^{19}$N and $^{21}$O 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
$^{19}$N$(-1p)^{18}$C$^*\!\rightarrow^{16}$C+$n$+$n$ is clearly dominated by
direct pair emission. The two-neutron correlation strength, the largest ever
observed, suggests the predominance of a $^{14}$C 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
$^{21}$O$(-1n)^{20}$O$^*\!\rightarrow^{18}$O+$n$+$n$, attributed to its
formation through the knockout of a deeply-bound neutron that breaks the
$^{16}$O core and reduces the number of pairs.
