EDIT: ah, maybe not. Let me amend that to "position is just rotated 180°".

Yes, there are at least 4 ways to obtain the type of positions we're talking about (from the perspective of at least one of the players):

- Have the players change seats
- Rotate the board 180°
- Reflect across vertical and horizontal axes
- Reflect across vertical axis and invert colors (of pieces and squares)

1 and 2 are obviously the same. Getting technical, 2 and 3 are equivalent because of a theorem in geometry which says all rotations are equal to two reflections, while 3 and 4 are equivalent because of chessboard geometry.

Geometry is definitely confusing and fascinating at the same time.

XPLAYER, they're not talking about if you reversed the colors of the pieces. Notice you've set up the black pieces as if they're white.

They're talking about a standard setup, but letting black move first instead of white. In such a case, the player to move first would have their king on the left.