Back and forth...

When I open the KDE Dolphin file manager I can use the keyboard short-cut F3 to divide the window and give me a right hand and a left hand copy of the same folders and files, with the one on the right active.

I can then close this Dolphin Split View using F3 again.

I can make the left or right copy the one that is active by clicking on either side with my mouse.

Is there a keyboard short-cut to move the active Dolphin window back and forth between the two halves of Split View?

You got me curious (mostly because, due to a little Carpel Tunnel Syndrome, I LIKE using keyboard shortcuts. So I did just a little digging by doing a search on Dolphin Shortcuts.
There's a lot of links available (like this -> http://www.shortcutworld.com/shortcu...le-Manager_1.7 ), but I wasn't able to find the answer to your specific question.

There's also a chance that the answer can be found under KDE shortcuts (I tried here -> https://docs.kde.org/stable5/en/kde-...shortcuts.html and other places). The most promising discussion about this, though, was here -> https://forum.kde.org/viewtopic.php?t=95924 , where they attempted to address assigning shortcuts and customizing. That may be the way to go, but I didn't read far enough to see if the question was resolved.

If you do find an answer, please let the world know!

Using the proved, logical method of discovery of pushing all the buttons, I found out I could use Tab and Shift+Tab to switch between two opened versions of a folder or file open in Dolphin created by pressing F3.

In KDE Plasma5, Tab or Shift+Tab cycles through the possible functions in the two opened versions, starting from right side version, which F3 opens as active.

Tab pressed once activates the slider at the bottom of the right side to enlarge that side's contents with the right/left keys. Tab pressed the next two times does something I haven't figured out, but pressing again selects the file tree box at the top of the left side and changes the background, showing the left version is active.

Press yet again and you have switched to the active left side presentation of files where you can work with them with the Dolphin file manager. Tab again is the slider for the left side, then Tab for a function I can't see, and the next Tab selects the file tree for the right and switches to the right.

The last Tab is the right side version of the opened folder or file you started from when you pushed Tab for the first time, ready to work on.

After using F3, hold Tab or push Tab five times to go from one set of files to the other, or Shift+Tab four times if you want to cycle in reverse.

This sequence is similar to the actions of Tab on an open folder or file in Dolphin, but I never could find a reference to any keyboard shortcuts for a double version of Dolphin.

The wheels of development sometimes run rather slowly :

https://bugs.kde.org/show_bug.cgi?id=171743