I think a 360 degree turn allows for those errors to be corrected as they turn with the aircraft whereas the Earth's magnetic field doesn't.
Exactly, but how does the system which is being rotated know the angle through which it is being rotated?
The fluxgate itself cannot tell the difference between (a) the field being bent on a particular bearing and (b) the speed of the rotation changing as you pass through that bearing.
But with a directional gyro, the system can work it out because it knows the relative heading.
Let's denote the 3D vector of the earth magnetic field with BM, and the local magnetic field generated by all aircraft objects (at the location of the magnetometer) with BL.
The projection of BM to the north-east plane points north.
When you rotate the aircraft around all three axes and record the measurement of the magnetometer, and have an aircraft with BL=0, you get a sphere around the origin, whose radius is the total magnitude of the earth magnetic field at the given point on the earth.
Now if you have a BL!=0, the magnetometer measures BL+BM. Thus, you get a sphere around BL, since BL rotates with the aircraft. Thus, the sphere fitting procedure I described above will find BL, and thus the calibration.
It doesn't work if you have local disturbances that do not rotate with the aircraft, though.
But see, you don't need to know how fast you rotate the aircraft.
That's a brilliant post, Tom.
Sphere fitting software... I recall fitting a polynomial to some points, but fitting a sphere is serious maths.
Now... where were you when we were discussing in-flight compass calibration using a GPS? 
I have just heard this from another pilot:
I think it’s brilliant
fluxgate compensation (that sounds very Star Trek like)
I thought this was ‘Back to the future’.
