Total Commander

I suggest to add the following commands to the context menu within the 'Synchronize Directories' dialog:
Select for moving -> (left to right)
Select for moving <- (right to left)

For single (non-duplicate) files only one direction is possible and the meaning is obvious.
This case is the main motivation for my suggestion.

For duplicate files both directions are possible and the logical meaning is that the destination will be overwritten (as with copy) and the source will be deleted.
Of course a real move is to be used if destination and source reside in the same partition.

As icons for the move operations I suggest to overlay the icons for the copy operations with a little red x indicating that 'move a b' is logically equivalent to 'copy a b, delete a'.
So the icons would roughly look as follows:
x->
<-x

Support++

As an alternative the following commands could be added to the context menu within the 'Synchronize Directories' dialog:
Move -> (left to right)
Move <- (right to left)

These commands would be applied immediately like for example the commands 'Delete left' and 'Delete right'.
With this approach no additional icons are needed.

On this occasion I would like to go one step further and suggest also the following commands:
Rename left
Rename right

These commands would be useful when you want to keep a duplicate file and rename one before copying or moving the other one.
In this case a pair of duplicate files would be turned into 2 singles.
The inverse case is possible too.