Question [Q] How to Rotate HDMI Output?

Image for starters:

I'm trying to display a Galaxy S4 (GT-i9500) on this computer monitor that I'm turning in to a "smart mirror".
I want the monitor in portrait mode. I need a way to have the phone output portrait mode sideways so that it fills the screen. Landscape fills the screen fine.
There are no settings in the monitor.
No rotation control apps will have any effect on this. They seem unrelated. Example, landscape/reverse landscape are the same exact thing on the TV. Same story on portrait.

Thanks for reading!