Is there a Chess GUI for Windows that shows both the board from white's perspective and black's perspective side by side, without the need for flipping?
The closest I've gotten to something like this is having separate accounts on whatever chess server you're using.
If you're using, say, FICS, you have two instances of your favorite client running, the first client is your main one and the second one you just join to the game through another account as a "watcher".
This requires a bit of manual labor, though. A more automated way is done through exploiting the macro functionality of most modern clients. You could, say, run the second client in the background at all times, and have the first client send an invite to the second to join the game, and the second client set to automatically accept invites coming from the first account.
At the very least, I think Winboard and BabasChess are capable of this. Others almost certainly are.
I doubt that there is such a chess program. You could probably accomplish this by opening the same game on two separate boards and flip one of them. The downside would be that you would have to browse in the two boards separately. So to move a move forward, you would have to click twice, once on each board. The upside would be that you can analyze different positions from the same game in parallel. Perhaps there is a setting to flip the board automatically according to who's turn it is?