I couldn't get Twitter to accept a PNG with a transparent background. I read that the Android client could do it, so I tried it there too and on my iPhone, and in every case the transparent part of the image would turn to white. 

I figured out how to get what I want anyway. I just created a page with the transparent image on the color background I want it look good on, then did a small screen capture, pasted it into the Mac Preview app, saved it and uploaded to Twitter, and it worked. 

It's a hack, hopefully temporary, but now I was able to get rid of the unsightly white splotches on my otherwise beautiful editor.