Apparently not because the handle is not a separate part and to put it upside down would mean the screw would show on top. I thought it should have been facing away on the left side of the toilet but its perfectly fine facing towards the toilet.
Dear Valued Home Depot Customer, We reached out to MFG for more information and below is their response. “ Hello, my name is Leah with KOHLER Customer Service. I am happy to assist you with your accessory question. I apologize you were unable to find the information you were looking for. The distance between the screw locations on the mounting plate for the Stillness towel ring is 25 mm. Please work with your local Home Depot for pricing and availability on selected product. Thank you for choosing the Bold Look of KOHLER. "
Yes, the hanger has that capability.