Why aren't recognizable languages and co-recognizable languages not reducible to each other?
Suppose that $L_1$ is recognizable, $L_2$ is co-recognizable, and there is a computable reduction from $L_1$ from $L_2$. The computable reduction shows that $L_1$ is also co-recognizable, and so decidable.

We conclude that if $L_1$ is recognizable but not decidable, then it cannot be reduced to any co-recognizable language.