Recognizability and complements

I'm learning about Turing Machines, decidability, and recognizability, and read that if a language is recognizable, its complement is sometimes recognizable. I don't really understand how this could be, can someone give me an example where a language has a recognizable complement and an unrecognizable complement? Thank you!

Example 3. The language $$H = \{ \langle T, x \rangle : T \mbox{ is a Turing Machine}, x \in \{0,1\}^*, T(x) \mbox{ halts}\}$$ is recognizable. In order to recognize $$H$$ it suffices to build a Turing machine $$M$$ that checks if $$T$$ is a valid description of a Turing machine, simulates $$T$$ on $$x$$ until $$T(x)$$ halts (possibly forever), and accepts.
The complement $$\overline{H}$$ of $$H$$, however, is not recognizable as if it was recognizable a Turing machine $$M'$$ for $$\overline{H}$$, together with $$M$$, would allow to solve the Halting problem. To do so simply simulate in parallel $$M$$ and $$M'$$ until one of them accepts.