Can the complement of an unrecognizable language be a recognizable language?
Sure: the Halting language (sometimes called $K$) is recursively enumerable whereas its complement ($\overline{K}$) is famously not.

Generally speaking, the existence of such languages follows from the implication you state -- the complement of enumerable yet undecidable languages can not be enumerable -- by the existence of precisely enumerable yet undecidable languages.

That said, there are also non-enumerable languages whose complement isn't either.