Clipboard only going from desktop to android, not android to desktop

Android doesn’t allow this automatically, not since A10 or so. You have to manually send to clipboard contents. You should have an option for this in your pull-down menus, or possibly via a persistent notification of you are not on a recent Android version.

You may be able to force it using adb but I am too lazy to bother, since the pull down works quickly for me, and I don’t want my Klipper saved history to fill up too quickly.