Is it possible to rearrange Grid Buttons?

Yes! -  Right Click on any Gird button to Cut and then Paste buttons to any unoccupied button within the Grid structure.
 
Cutting and Pasting a "Grid" button does recreate the entire structure of a multi-layered grid button in the the new position.
 
 
Is it possible to rearrange Grid Buttons?
1

Right Click to Cut a button out
of the current grid position

1. Right Click to Cut a button out
of the current grid position
To relocate any button within the Grid system, Right Click on the button and then choose Cut.
Is it possible to rearrange Grid Buttons?

Select Cut to remove the button from the current position

1. Select Cut to remove the button from the current position
Any type of button may be moved, Item or Grid.
2

With the Cut completed,
Right Click in the new position and
then select Paste to relocate button

2. With the Cut completed, 
Right Click in the new position and 
then select Paste to relocate button
Navigate to any unoccupied button within the Grid structure once a button is "Cut".  Right clicking on an empty button will then allow the user to Paste the entire contents of the button into the new position.  
 
Is it possible to rearrange Grid Buttons?

Select Paste relocate the Cut button into the new position

1. Select Paste relocate the Cut button into the new position
Selecting the Past button will reassemble the contents of the button, Item or Grid, into the new position.
 
Is it possible to rearrange Grid Buttons?

New location for the selected button in this example

1. New location for the selected button in this example