As has been discussed before (and elsewhere) there's no difference in the analysis provided through "flip-and-diff" or "rotate-and-diff" since you're only looking at the horizontal axis of the mirror. Lack of symmetry only shows itself in the "rotate-and-diff" operation because of test stand astigmatism (primary contributor); displacement of the source and KE (not a problem in most setups where they are within a cm or so, ideally half a cm; and lastly actual astigmatism on the mirror itself (which Foucault is completely blind to).
So it's not even an academic question, the method described in this thread ("unmasked Foucault") is about extracting fine grained readings across the horizontal axis of the mirror, and secondly doing some local averaging to pin down those diameters to subpixel accuracy (necessary for the process to represent the mirror axis faithfully). Local averaging would be done across slightly tilted diameters of the original data (in essence accomplishing the "rotate-and-diff" directly). Over small angles the symmetry of the mirror is more than sufficient (actual tests of actual mirrors) to support either method.
This thread is not really about producing full surface mappings, as that is not what Foucault lends itself to or can be applied to, IMHO. For that you really want to explore alternative techniques, Hartmann testing, Shack-Hartmann imaging, and IF, all of which measure the mirror surface contours either directly (IF through OPD differences, Hartmann variants through measurement of surface tilt). Foucault measures slope.
Since we've been discussing this via PM for some time now I have some idea of what you're trying to do, but if it's going to become long and complex I would really like you do it some other thread, one devoted to the method you're trying to propose here. This thread is devoted solely to unmasked Foucault testing as described above and in the introduction, and I've got enough experience with it now to appreciate the immense value of an assumption free, ancillary optic free, and most particularly bias free method of very accurately acquiring slope data for mirrors.
Just to be completely clear - Foucault only measures the radii of zones across the mirror which null at the same longitudinal position along the caustic curve. That's ALL it does. The information may look like it can be processed independently without regard to the nulls on both sides, and you're more than welcome to pursue that idea, but I request specifically that you do that in some other thread as I consider it off topic here.
I concede it is remotely possible that Foucault testing has been used for 150 years with nobody ever realizing that it contained a hidden method to extract accurate full surface mapping comparable to, say, IF - and that of course would be a welcome discovery. I don't think that's very likely though - and with respect I don't think you've discovered it just by working over a few Foucaultgrams - and it doesn't stop me from considering it in detail, certainly, as there may be new life in the old test yet.
Now, to get to what I hope will be the end of this discussion here- Foucault has, necessarily, different sensitivity at different portions of the caustic. The closer you get to the paraxial focus (of the center) the less the sensitivity becomes, until ultimately you have none at all (contains test stand astig and in-process roughness) at the paraxial focus itself:
There is no necessary condition of the center line of the mirror that can return data on a null that's accurate enough to use to determine the radii of either side independently of the diameter you're actually measuring. Just take a stab at this rotate-and-diff image of a 55% complete 18" f/3.58 mirror, showing some test stand astig under the rotate-and-diff (not in the mirror to be sure):
If you measure from the left null to the center line, and from the center line to the right null, you get different numbers. But if you measure from the left null to the left edge of the mirror, and from the right null to the right edge of the mirror (all here well centered) you get the same numbers. This demonstrates clearly that the nulls are where they should be on a mirror with good figure of revolution, but that the center line is ill-defined. The center line HAS to be there, somewhere, in this test method, as somewher between the left side and the right side there is indeed an area with the same values under rotate-and-diff (because the curve across the face graduates from light to dark to light to dark as it should) but the information in that graduation is only useful where the change is steepest, as illustrated below in a composite picture showing the actual overlay:
Note that the nulls on the outside lie clearly on the crest of the waves, as it were. But the outer crests are clearly delineated, as they are actually crossing the axis , where the KE always is, at this particular longitudinal (4.25 mm from paraxial focus). The center OTOH has no such delineation - it is very far from any focus at all, and as such it is at the mercy of minor diffraction defects in the image, small variations in illumination, sensor response, and the like. This is a critical point, because the sensitivity of the Foucault test depends strongly on this delineation. Where you don't have that critical zonal crossing you just don't have good data, and you absolutely need good data to get to subpixel accuracy in running this particular test, or you will get garbage out.
It occurs to me you might want to look at producing an automated version of the traditional Caustic test, where the zones are nulled at their actual foci along the caustic horn, and separated, not joined along the center longitudinal line. That test relies on measuring the actual separation of the two separate focal points at given longitudinals, and so could directly tell you about the two sides of the mirror. It can be done with a wire and an unmasked mirror, but it will require automation in x and z to a very high precision, and orthogonality, to yield useful data. Unmasked Foucault currently appears to achieve the same sort of precision (1/100th wave or so) without elaborate mechanisms, but sacrifices knowledge of the full surface in order to produce an average of the particular axis measured.