It is possible but is not recommended because
only treating one arch can result in bite issues. We not only align the teeth
but also give all of our patients a stable bite by treating both the upper and
lower arch.
From your description your bite sound asymmetric; ie, different on the right than on the left. In a case like that headgear may not be the best choice since it corrects both sides equally. This means that it would push your class I side into a class III while correcting your class II...
I have yet to see a case where a patient ended up with hollow cheeks from Invisalign Treatment. In fact, the opposite is usually true since Invisalign expands/broadens the arches -- it may result in fuller lips or cheeks.