Does ptosis also cause face asymmetry? (Photo)

I would recommend you have a review with an experienced oculoplastic surgeon to assess the ptosis, evaluate the squint and eyeball (globe) position. 
If the levator function is good, the pheylephrine test is an excellent method of visualising the potential surgical outcome, before actually undertaking any revision surgery.

Your upper eyelid is ptotic, causing facial asymmetry. Fortunately, this is correctable with several different types of surgical procefures.  See an experienced oculoplastic surgeon who will evaluate your condition and guide you with appropriate treatment. Good luck.

There are several disorders of facial asymmetry that also include ptosis, however, not all forms of ptosis are accompanied by a truly asymmetric face.  A droopy lid can give the impression that that face is asymmetric as the lid is in a different position and is often accompanied by an elevated brow, a compensatory response to a droopy lid.  I recommend having an oculoplastic surgeon or lid specialist exam you before a revision surgery is performed.

Although your eyelids are certainly asymmetric, most true facial asymmetries are related to the bone.  Your eyelid certainly needs to be repaired.  If what you currently have is the result of an attempted repair, your surgeon owes you an explanation or you need to find another surgeon!  Not all oculoplastic surgeons can do this repair nor can all plastic surgeons do this repair.  You need to find one that has a lot of experience with the surgery and likely interveiw a few of them to find the right fit.

It appears you have left sided upper eyelid ptosis and since your right side appears normal this is causing asymmetry. It's possible to repair this problem.  If you are serious about addressing this problem you should consider seeing a board certified plastic surgeon to discuss your options and find out if you are a good candidate for treatment.