Yes, it is possible to print both sides of the headphones differently. After uploading your image you will get the following message:
"Would you like to use this setting for the other side as well?"
Click on "Select another image" to select a different image.
You can delete the image for the left side by clicking on the "trash icon".
Click on "Edit right side" to upload an image for the right side, if you don't want to print the right side, just click on "DONE".