It is well known that deterministic Büchi automata (DBA) are less expressive than non-deterministic Büchi automata (NBA), and in particular DBA are not enough to cover linear temporal logic (LTL). However, the argument often used is that DBA are not closed under complementation - that is there is a language $\mathcal D$ recognized by DBA, but $\mathcal D^c$ is not recognized by any DBA. As an example, $\mathcal D = (b^*a)^\omega$ and $\mathcal D^c = (a+b)^*a^\omega$.
I wonder though, whether for any $\omega$-regular language $\mathcal L$ it holds either $\mathcal L$ is recognized by some DBA, or $\mathcal L^c$ is recognized by some DBA. If not, what is an example of a language which is not recognizable by DBA and which complement is not recognizable either?