When using a 360 head (whether the new 360 Head Maker or the Campus Hero ones, but 360 heads in general) with an Animation Motion, the head—no matter if I make it face in another direction using the Face Key Editor—does the animation if it was staring at the 4th wall instead of the other characters/whatever I was having them look at.
Even if I use the Face Puppet tool—while in the preview, it looks like the head will stay in the place I set it with the Face Key Editor—when it records the motions, the 360 head faces the 4th wall/camera/audience and makes the motions.

What am I doing wrong? I seem to be in the minority here since I keep running into issues barely anyone else talks about. 

When you apply an existing motion file to your character, whether it be for the face or the body it's always applied exactly as the key frames were saved when the motion was created. For face motions if you want the head to be turned, apply all the face motions you intend to use first then go back through with the Face Key editor and add in all your head turns etc.

- TET
Animation and Video Life
The Lazy Animator - Tutorials for Cartoon Animator

From my testing, you have to check the option "Blend data on next recording" that will keep your Face Key Editor keyframe and blend to record facial animation.