How to change rotation axes in quaternion?

Question

Mathematics

I have an inertial sensor that outputs quaterions and an acceleration vector.
The controller is oriented as
Z - in gravity, X - "right" in the direction of view, Y - in the direction of view.
And the bottom line is that when I twist along the X axis, that is, I try to look up and down, then everything is ok. But the other axes are reversed. That is, if I spin the controller in Z, then in the game space it spins in Y and vice versa.

My goal is to swap rotation axes in a quaternion. How?

Answer 1

Suddenly, when I decided to indulge in permutations of variables in my code, everything was decided.
In my case, it was enough to swap the W and X components, and then invert X. (For me it rotated in the opposite direction)
That is, in the end I did (w, x, y, z) -> (x, -w, y, z)
I also googled someone else's code and found another permutation (w, x, y, z) ->(-y, x, z, w), but in my case it did not give the desired axes.
Some permutations completely break the engine and the space starts to collapse, so be careful