Hello, I am currently trying to rig a cartoon mouth using shape keys. With my character, the mouth can appear on either side of their face. The only problem I have is that when I create a new shape key with a flipped mouth position, the whole mesh flips in addition to having the normals flipped. 
Here is an example:
Can you provide some examples how would you like mouth to move? I mean some cartoon videos.
Right now i can guess that you need more like a weight painted vertices attached to a bones, than a shape keys.
Hmm… This is the closest thing I could find.
(Skip to 0:25)
You’ll notice that his mouth switches sides on his face.
I assume that you got the inverted shapekey by mirroring the vertices on the x axis, this doesn’t work as you can see.
You could create a duplicate of the shape-key you want to mirror, symmetrize it and then restore the side you don’t need by the ‘Blend from Shape’ command (that is, select the vertices and run the command with ‘basis’ as source and ‘add’ disabled).
Just select vertices around mouth, hit G with proportional editing enabled and move mouth wherever you want. Don’t mirror anything. Make a shape key from that move.
I appreciate all of your help! I am not very familiar with the ‘Blend from Shape’ command. Could you go into a little more detail about the process you described please?
‘Blend from Shape’ is a command that allows to blend the displacement of selected vertices in the active shape-key with that of another shape-key of the object.
So, if you want to either substitute or merge some deformation from a shape-key to the current one, you have to select the vertices to be affected, making sure the shape-key to edit is the active one in the Object Data panel, then run the ‘Blend from Shape’ command from the menu: Mesh -> Vertices.
From the Undo palette (F6 key) you have to choose the source shape-key you want to merge from, and its amount; if the ‘Add’ option is disabled the current dislocation of the vertices on the active shape-key will be substituted by those of the source key, while the ‘Add’ option enabled will add the source dislocation to the existing one.
On the other hand if you need to restore some vertices to the original position, you will run the command with ‘Basis’ as source, strenght = 1.0 and ‘Add’ deactivated.