Why aren't recognizable languages and co-recognizable languages not reducible to each other? While learning to prove undecidability of problems, I came across a statement that you can't reduce a recognizable language to a co-recognizable language and vice-versa to prove undecidability. Why is it so?

Eg: Halting problem of TMs can't be reduced to Emptiness problem of TM

$\qquad$ $H_{TM}$ = {$\langle M,x \rangle$|$M$ is a TM and $M$ halts on input $x$}

$\qquad$ $E_{TM}$ = {$\langle M \rangle$|$M$ is a TM and $L(M)= \emptyset$}
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.