Hey, Holgy, unlike some other people in this thread, I understand your problem, and did not assume you are a moron who doesn’t know how to turn off the portrait lock.
I have the same problem. When playing a music video from my iPhone to Apple TV, only the sound is played. And, you can’t watch videos in landscape. When you turn the phone, you get a sea of album covers instead of your video.
WORKAROUND
I was able to get around both problems by playing music videos with the Videos app – not the Music app. Music videos played in landscape on the phone. When I used AirPlay the AppleTV showed video, not just audio.
The drawback is that the Videos app is poorly organized. It is sorted only by song title, not by artist or album. It does not have a search feature that I could find, nor does it have the alphabet selector down the right side of the screen. This is fine if you have only a few videos, but if you have a lot, it would be a mess.
LANDSCAPE MODE IN THE MUSIC APP
Other people complained about not being able to play music videos in landscape (from the Music app). Apple says the feature is no longer available. See response by “bluebearies” in this thread.