If A many-one reduces to B, does the complement of A many-one reduce to the complement of B?
Let's try an example. Here is a reduction from the set of even numbers to the set of squares: $x \mapsto 2^x$ (check it!). It is also a reduction from the set of odd numbers to the set of non-squares. Is that just by chance?

We can make the following twist: the mapping $x \mapsto 2^{x+1}$ maps the set of even numbers to the set of non-squares. Is it always the case that we can modify a reduction from $A$ to $B$ to a reduction from $A$ to $\overline{B}$?