Do you know the saying Only Nixon could go to China? I didn't 'til just now. It refers to the 1972 visit and as a metaphor means that somebody (or an organization) that has a reputation to strongly lean one way, has much more leeway to act in the opposite direction than someone else.

A few examples that come to mind immediately:

• If a (European) Green Party were to turn pro-nuclear for climate-change reasons, after decades of opposition, they would immediately have more credibility than the parties that always have been for nuclear power.
• It took the social-democrats to dismantle the western European welfare states. Similar attempts from the right would have been fodder for the next election.
• Super-rich people that want higher taxes.
• Were the Swedish Moderates, who opened up schools to the private market in the 90s, to admit that this was a bad idea, the perpetual stories about horrendous consequences might actually have something done about them.

Interestingly, politicians usually don't want to be "caught" changing their mind on some issue, which might not be the optimal strategy in light of the above.

And it should be very effective for someone with an agenda to try joining the other side and change their stance from within; the low chances of success might be outweighed by the larger impact in that case. The episode of The West Wing about a gay Republican (in the early 2000s) fits that idea.

I think Nixon in China is one of those mental models that is worth keeping in the back of one's mind for a while and see how many more examples show up.