(1) I wouldn't have designed it the way they did.
(2) *They wouldn't have designed it the way they did.
I marked the second one as ungrammatical, because, well, it doesn't seem like it is grammatical to me. And yet I can't put my finger on whether I really think it's a syntactic or semantic problem. It feels very related to the difference between
(3) He hit him.
(4) He hit himself.
in that everything hinges on whether the pronouns are the same or different, but both (3) and (4) sound fine. (3) is one person hitting another, (4) is someone self-hitting. (1), though, is a statement about one person disagreeing with a group, and (2)... just doesn't make sense by itself. Extend it with an appropriate counterfactual complement, however, like
(2b) They wouldn't have designed it the way they did if they knew how it was going to turn out.
and it sounds fine.