If you give ground when you are right it makes it seem like maybe you are wrong to people who haven't made up their minds, it vindicates people who are wrong and it makes you look weak.
(This also kind of works when you are wrong... but *think* you are right... or if you are very good at lying. Which complicates it a bit.)
Nonetheless, it's not possible to have one set of stances for one part of the public and a different one for another. Everyone will find out.