Both sides are totally opposed, yet see the same fact as proving they are right.

If redheads are 10 times more likely to be in jail for violent crimes, it is evidence for both "redheads are violent" and "judges hate redheads" - and both might be true!

And "redheads are violent" and "judges hate redheads" are not totally opposed, they only look that way in a context where they are taken as arguments in support of broader ideologies who, them, are totally opposed (or rather, compete with each other so oppose each other).

More generally, many facts can be interpreted different ways, and if one interpretation is more favorable to one ideological side, that side will use that interpretation as an argument. Seen like that, facts looking like they support "opposite sides" seems almost inevitable.