Hi Jerry,
I would have to ask if you are having problems after your flip? If not, then I think you have your answer. See below about PHD being able to do a mathmatical inversion of the model.
I always calibrate in the frame of my target, typically after centering for the first time. First, I believe that it is best to calibrate within the field that is being recorded anyway. Someone can correct me if I am wrong, but when you see the calibration process working, it makes perfectly good sense that this is best. Only exception is that I know that I have failed to get PHD Guiding to calibrate close to the pole, but I am not sure going to the the ecliptic is going to help in that situation. Luckily, I have gotten decent guiding even in the event of a failed calibration near the pole.
Second, if you look at the calibration model, it becomes clear that the math during a flip to invert the model is pretty trivial. So PHD Guiding does not normally even ask to reboot the calibration in that case. The problem is that some peoples' mounts have issues in the flip. That can be evident after the mount flips and the frame rotation is not a true 180 degrees inversion or close to 180. If you have that as an issue, by experience, you should determine whether a recalibration really offers any real improvement in your guiding. I wouldn't waste time redoing a calibration unless it is required for your situation. Always best not to fix, which is not broken rule.
So how can a mount not get a perfect 180 inversion between flips? Beside thinking that a polar alignment is perfect, I think that most people forget that you can get a good polar alignment even if you mess up the level of the mount. In such a case, an unlevel mount means that the polar alignment only looks good. People who have a permanent pier typically don't have to worry about that! Except once. I just saw a post regarding poor flip numbers because the mount was poor. So not all mounts are perfectly orthogonal in construction. That is one reason, other than level issues. In my case, I get good guiding without recalibration through the flip. And my AM5, which has two bubble levels is the only way I level my mount. With my bad weather and short windows of opportunity, I have no time to fool with perfect level, and the crappy levels that are on my AM5 do not even agree with one another! Like, which one am I supposed to believe. Given that situation, I know that one level is wrong, and that is the best case scenario. The other scenario is both levels are wrong! I know that my levels are off because when I do my polar alignment, if perfectly level, adjusting Dec should have zero effect on effect on Az and visa versa. That never happens for me, so it can only be level being off. But it is never that bad, so I let PHD do the job of corrections during the imaging. Thats what I pay it for!