Is there a way to see the device's screen *live* on PC, through ADB? I am a vlogger and after getting HTC Desire HD, found out its camera was all I ever needed. However, I usually have to see myself while recording video, to control my facial expressions, and with a rear-facing camera it's just not possible.

I know ADB allows for screenshots, but is there a way to stream the screen output to my PC's display? 
Not necessarily at 20+ fps, just a couple frames per second will do.
screenrecord
screenrecord is an internal Android executable that dumps screen to a file, and ffplay from ffmpeg happens to be able to play an H.264 encoded stream from stdin
First enable ADB USB access and then on Ubuntu:
sudo apt-get install adb ffmpeg    
adb exec-out screenrecord --output-format=h264 - |
   ffplay -framerate 60 -probesize 32 -sync video  -

You might have to make the screen move a bit before you see anything:

Uncut demo: https://www.youtube.com/watch?v=fVgeoMYm61Q
Explanation of parameters:

-probesize 32 -framerate 60 -sync video reduce the delay between the image on the phone and the image on the disktop (thanks to @arbuz)
we use adb exec-out instead of adb shell because shell might mess up some control characters due to being a shell (thanks to @Bowi)

See also: Use adb screenrecord command to mirror Android screen to PC via USB

Tested on Ubuntu 20.04, Android 11, Pixel 3a.