I was just writing another thread when I stumbled upon something strange:

What if I now make a slight change and replace ##\bf{b}\rightarrow-\bf{b}##? The expectation values containing ##\bf{b}## change signs and I get
$$\left|\frac{1}{\sqrt{2}}-0\right|=\frac{1}{\sqrt{2}}\leq 1+\frac{1}{\sqrt{2}}$$
which is correct! But I've only turned my measurement device 180°, i.e. exchanged the (+1) and (-1) results, or more mathematically, I've permuted my measurement basis. How can this lead to such a profound change, i.e. this would allow a local-realistic description for these measurement angles? Surely there has to be something wrong.

Yes I know that, but in this case I'm not even really changing the configuration but only swapping the up/down-outcomes. The measurement basis is the same as before (only permuted), that confuses me.

no your b vector is rotated by 180 degrees so the angles are not the same any more. Usually Bell inequalities are violated when the angles between them is 45 degrees.

You don't change +/- outcome, those are always given by probability 1/2 1/2 by QM.

But it's not because you can simulate the result by hidden variables that nature will do it. How will it know the configuration permits to do that ?

This is not correct. There are combinations of angles, usually 3 or more, that lead to an inequality. Not all combinations do have that feature. 45 degrees can be used with other specific ones.

I'm still confused. ##\vec{\sigma_2} \cdot\bf{b}## and ##-\vec{\sigma_2} \cdot\bf{b}## have exactly the same eigenvectors. This means the measurement statistics (probabilities) are exactly the same, and so are the post-measurement states. Only the results (+1) and (-1) are swapped. I still don't see how this is "significant enough" such that the Bell inequality is suddenly satisfied.