On the right on the breadcrumb bar is the button "show folder in the other file display". But this button copy the tab instead of moving it.
Is there a setting to change this? I could not find anything related to my question, but I do not really see the sense of copy the same tab to the other side... But move I use sometimes (possible via context menu and drag&drop, yes)
You can drag tabs from one side to the other to move them over. There are also options in the right-click menu for tabs. Or you could make a hotkey or toolbar button to do it if you prefer.
It's common to want to see two nearby folders in each side, so getting the current folder as a starting point in the other side is something people do a lot. Less common, but still useful sometimes, is viewing the same folder in both thumbnails and details mode.
ok. Thank you.
I recognized that I could add a new button directly next to the existing one with this command:
Go TABMOVE=active
Set SOURCE=Toggle
The tab is moved and has the focus. 
Exactly what I want. So I have no both options next to each other.
(Source: Modify R-click tab "Move Tab to -- (L/R)" but then "Activate" tab?)
@Leo But one hint regarding button creation, that was confusing: I was only able to place the button on the left side from the path field in the breadcrumb bar, but after accepting the button was placed on the right side. When I use customize the button is shown again on the left side.
Is there also an option to define different symbols depening on the lister side like for the default button? Sure, only a cosmetic topic due to the arrow direction...
You can turn off the special rightmost buttons (Preferences / File Displays / Border), and replace them with toolbar buttons that do the same things in the same place. Then you can move other custom buttons among them.
The Go FDBBUTTONS command lets you add some or all buttons to specific places on a toolbar.
e.g. Go FDBBUTTONS=copy,layout added to the end of the toolbar will put Copy and Layout buttons there.
Ok. The Button is placed correctly for me. Only the behavior during customizing is "confusing", because it changes the position during customizing.
What I meant: the button for "move folder to other file display" shows an arrow to the right in the left display and an arrow to the left in the right display.
But I guess such a Button symbol depending on the display side can only be defined by you.
If I may add my input. It's typical for me to want to control what is extracted from a zip or .tgz (aka .tar.gz) file. To that end, I'll use that button so I have the Downloads folder loaded on both sides, create a new directory for the extract on one side, and then, back on the other side, double-click into the extract and pick out what I want to be copied.
I know I could use an extra, permanent tab with Downloads showing, but it's roughly equivalent to go up to the "Display this folder in the other file display" button or down to a tab.