I have been on the hunt for what exact shortcut the controller hides from us. But there is a steam menu (controllers, advance, show more, at the button) that shows them!
Quickly want to change your volume? Try steam key + left joystick!
Sadly you can not change anything here, altought the menu shows you you can, nothing will be saved after you leave a menu. (I'm not a fan to put the left mouse button on the right button)
A few of them I manually found before written out, also for the people without images:
hold A + R1 + Steam
To use Puck wireless mode right
hold A + L1 + Steam
Start Puck wireless mode left
hold B + R1 + Steam
Start Bluetooth mode
When on (solid white)
Steam+Y
Turn off quickly
Hol steam (10s)
Turn off
Steam + left joystick up/down
Change volume level
R4 + R5 + L4 + L5 + Steam
Backpack mode (ignore buttons until this combo is used again or USB)
Steam + Right trackpad/joystick
Mouse movement
Steam + X
On screen keyboard
Steam + R1
Take screenshot
Steam + Menu
ALT+TAB
Steam + D-pad right
Enter key
Lizard mode?
How the controller works without Steam running.
I hope on Linux it works (or will work) the same as the Steam Deck's controller. There you can switch to gamepad mode by holding down the start button for 3 seconds or so.