“Infinities are weird” might satisfy your criteria.

It surfaces in such things as

- why do we consider 0.9̅ and 1 to represent the same number?

- why are there ‘as many’ even integers as there are integers, and ‘as many’ integers as there are natural numbers?

- why are not all infinites equal?

If you don’t know/accept these, you’ll keep using intuition that served you well for many, many problems on finite sets on problems involving infinite sets, with disastrous results.

I've heard compactness described as one such concept.

https://math.stackexchange.com/questions/485822/why-is-compa...

I can basically confirm all of those. The delta-epsilon definition of a limit was what really kick-started my interest in mathematics, but it was also so brutal to wrap my head around.

I would say the concept of a group is another one for a lot of people. Abstract algebra is sometimes the first time you take a math class in undergrad where all references to numbers are very far away, and you have to accept that the definition of a group contains vast and untold multitudes beyond its straightforward definition. When it "clicked" for me, however, everything else in that class started flowing much better.

Gauge invariance in mathematical physics might be another good one, but that's a lot more niche in who actually learns it. Lyapunov stability from dynamical systems is something I'd point out as actually the opposite of this - it makes a lot of sense very quickly and helps make light of quite a few things you see in the dynamical systems course material leading up to it.

Probability theory. It is closely aligned to my intuition now, but when I first learned it, it was difficult to accept beyond manipulating formulae.

Linearity aka most of linear algebra. Again, beyond manipulating formulae, many concepts eventually become intuitive with enough application, but its a hard won intuition to acquire.