One should keep in mind that both parties were "big tents" at times in their history. For example, Nelson Rockefeller and Barry Goldwater were both Republicans at the same time while Hubert Humphrey and Strom Thurmond were both Democrats at the same time.
I mean, that is an indisputable fact. However, I think people use that "fact" to fit their agendas, and they end up saying something like Nelson Rockefeller was more liberal than Robert Byrd because of civil rights alone, which is absurd.
People certainly do take certain aspects of the "tents" out of context to paint a picture that suits their agenda. They'll often pretend that the aspect of the "tents" that don't suit their agenda never existed. I'm also aware that "conservative" and "liberal" are far more nuanced than civil rights.