$NL^2 = NDSPACE(\log^2n)$ is closed under complement From Savitch's theorem we have $NL^2 \subseteq L^4$, which is deterministic and thus closed under complement.
From Immerman–Szelepcsényi theorem we have $NL = coNL$.


Why then $NL^2 = coNL^2$
The Immerman–Szelepcsényi theorem states that $\mathsf{NSPACE}(s(n))$ is closed under complementation whenever $s(n) \geq \log n$.