undecidable problem and its negation is undecidable
Consider the following language:

$$L_2 = \{(M_1,x_1,M_2,x_2) : \text{$M_1$ halts on input $x_1$ and $M_2$ doesn't halt on input $x_2$}\}.$$

$L_2$ is undecidable and not semi-decidable, and same is true of its complement.  Why?  The intuition is "$M_2$ doesn't halt on input $x_2$" isn't semi-decidable, so $L_2$ is not semi-decidable; and when you look at the complement of $L_2$, the same thing happens for $M_1$.  This can be formalized more carefully using reductions.

More generally, if $L$ is a language that is undecidable and not semi-decidable, then

$$L' = \{(x,y) : x \in L, y \notin L\}$$

meets your requirements: $L'$ is undecidable and not semi-decidable, and the same is true of the complement of $L'$.