Try moving the device in figure 8 motions in a horizontal plane. That should reset the sensor properly.

The "calibration" didn't help. Besides, it makes no sense.

The compass on my iPhone works just fine as long as it's roughly horizontal. There is no such thing as "calibration" for the iPhone.

What's the difference between the two compasses? If they both sense the Earth's magnetic field, then they should both stay pointed approximately north regardless of any motion.

I took the iPad outside for better GPS reception, and eventually the Compass app started to work properly, sort of. I know that some iPhones sense the Earth's magnetic field via the Hall effect. I guess my iPhone, but not my iPad, has a Hall effect sensor. The iPad must sense the direction of the GPS signals and must know where the satellites are. That's a more complicated and less reliable way to tell where north is. I still don't understand how the calibration works, but it seems to be necessary.

Go to:
Settings - Privacy - Location Services - System Services:
and turn off Compass Calibration.

Then the compass started working properly.
It's nuts I know but the compass refused to indicate north until I did this.

It didn't work for me.