Without seeing your puppet, the "Head" puppet will rotate around its origin. If you don't have an origin handle, then you'll use a default handle in the center of the mesh.
Selecting the "Head" puppet in the Puppet Panel should show you handles. I'm guessing there's an origin handle right where it's rotating. The solution would be to move it.
I see a few things here.
You have two handles named 'Hair' perhaps because you have two layers named 'Hair'.
When you import artwork into Character, it will create a handle for each layer and place it at the center of your artwork.
You can see that most clearly for 'Glasses' and 'Face' handles which appear at the center of the corresponding artwork.
One of the the two 'Hair' handles also seems to be placed correctly but the other, the one outside the face, seems wrong.
This could be due to same stray/accidentally placed points, brushes in your artwork. You may want to track those down within another layer named 'Hair'.
The same stray artwork may also be displacing the 'Head' handle, which is presumably a parent layer of the stray 'Hair' layer that pulls the center of the artwork to the right.
Cleaning up the stray 'Hair' layer should move the 'Head' handle back to the left. You could also have manual control over that 'Head' handle. Placing the origin guide in your 'Head' will move the 'Head' handle to that location. The head will pivot around the 'Head' handle so by moving it around you can control the pivot point.
What happened there must have been when I dragged the layer named 'Hair' from the sample 'freda.psd' file into my artwork, because I couldn't actually see anything on the layer, I didn't really know what it was at the time.
I did the same for the layers named 'origin' too which are appearing in the top left corner.
Maybe it's better to not include these in my PSD if I can add/move handles directly in Character Animator?
I'll start again from scratch and check some more of the tutorials.
From my experience of CA so far I think it'd be nice to include a loop function in the future, it's probably not needed for most character animations, but I'm making some looping GIFs for the new website at the moment and unfortunately I had to sacrifice using the head rotation and position modifiers to make a reasonable looping graphic:
I don't do much character animation, but I'm favouring the manual way of achieving this look at the moment, CA might grow on me if I give it a chance. Here's one I did just using the Puppet Tool and Liquify effect in AE for comparison:
Thanks for your help guys.
Looping is on our list as is allowing everyone to create these and other behaviors.