From my past experience with BoundingBox XYZ, I believe the Min Max point of the PickedBox is correct.
Just like the BoundingBoxXYZ, the default value does not depend on the view direction but it conforms to the BasisX , BasisY, BasisZ of the standard coordinate system. Unfortunately, the PickedBox does not have the transform property like BoundingBoxXYZ, thus, you can not transform it to your desired coordinate.
I also believe this is quite consistent to what Autodesk has been doing with their internal unit/ coordinate and user units/ coordinate in other product and API such as AutoCAD (WCS and UCS). Besides, when you use API, what you want to know is what are inside the box and the location of the box, which can be relative Min point and Max point to the standard coordinate.
As for @RPTHOMAS108 's reply, I found that the directional style only differentiate the way to determine if an element is inside the box or not (left - right: completely inside, right-left partially inside, which is consistent with user action on Revit), but not the coordinate defining Min point/ Max point.
The way I see it, @KM_Yotsuha must have performed the pick box operation on a floor plan view, whose elevation is at 0, and the BasisY points to leftward, BasisX points upward instead of a standard coordinate where BasisX points rightward and BasisY points upward. Adding some view translation/ rotation to consideration, I thinking the result makes sense.
This is how you pick

This is how Revit interpret the picked box

I could be wrong, though. Looking forward to feedback from you guys.
P/s: I have edited the second images for better understanding.