Many places use a Mapkey to redefine the views so it's a 10 second process. Views are typically defined parametrically; if they are defined relative to the datum planes, the datum planes will appear in the same orientations as before; the solid geometry will appear to move, so that what appeared to be the top view of the part will result in a bottom view of the mirror - which is exactly what mirrors do.
The alternative to reversing the sign of the Z coordinate to create a mirror part is to use floating point transformations which can result in features failing to regenerate in the mirror that successfully worked in the original part; even if they don't fail they can be distorted by round-off or truncation errors.
I'm still missing what 'upside down and rotated 180 degrees' means to you. In this picture you say Z is rotated 180 degrees, but Z is an axis and cannot rotate. If it was mirrored, then the part would have a left-hand coordinate system, which AFAIK PTC doesn't support. It looks like the part is rotated 180 degrees about Y in the image, but the colors chosen blend too much to make out whatever detail you are emphasizing. Rotate the view around Y and the Z axis will point up.
It would help if the comparison images were from similar points of view, and similar scales and not have overlapping parts. I really can't tell what the difference is from zooming in on a CSYS that is the same size no matter how zoomed up it is. The above image cropped off the main feature that I use to tell if it is the mirror part and what the orientation is.
It would also help if you included a picture of the feature definition for all the new coordinate systems you made to show what the transformation definition was. When you show you are using a new one, it doesn't show that it was defined correctly. By default a new CSYS will have the exact same orientation and location as the default or previously selected CSYS. I suspect that's why it appeared to work one time and not another; it actually worked the same, but the definitions were different so the results looked different.
