He was holding his phone the wrong way. Just means 'incorrectly. eg maybe he was holding it between his knees.
He was holding his phone the wrong way around. Means 'reversed'. Sounds like he had the mouthpiece against his ear, and the earpiece against his mouth.
I'm having similar problem in the sentence below. Would you help me Clive. I don't know if headed paper means what trying to say. business paper is the same?
I printed the wrong way around on the headed paper. I put the paper the wrong way in printing machine and the text is upside down,