I recall that circular polarization in the QPH-031 is achieved by delaying one linear polarization with respect to the other and then combining them. One combination results in left circular and the other right. One can achieve the same externally with a fairly inexpensive solution, but there is one nasty complication and a few minor headaches.
If you want a universal LNBF that can do all four polarizations, it will be easier to start with a universal linear LNBF than with a QPH-031. If you have to ask why, please move on. Invacom makes such a unit, model QTH-031. This has four fixed outputs, which likely has advantages for this application over a four switched outputs. It also costs less.
I expect the QPH-031 performs some form of delay and combining at the original RF, which is 12.2-12.7 GHz for circular Ku-band with this unit. If Invacom implemented a constant time delay across the passband with a perfect 90 degree phase shift at 12.45 GHz (center), signals at the band edge would not be delayed by exactly 90 degrees. However the error would be bounded by about +/- 7 degrees, which will still lead to an acceptable combining. Invacom may have been even craftier and implemented a more constant phase shift filter; I have not peeked inside.
With an unmodified QTH-031 we would need to do the delay and combining at 950-2150 MHz. If we choose a constant time delay that produces a phase shift of 90 degrees at band center, we will be off +/- 140 degrees at the band edges. That won't work. I assume those still reading do not know how/want to design a constant phase delay filter over this range, so I will have to propose another solution.
Before proceeding, I will assume the linear outputs of the QTH-031 are coherently derived from a single oscillator for each of the two bands. It wouldn't make much economic sense otherwise, but I have not checked this premise. If my assumption is wrong, the following approach will never work. A 90 degree phase shift will require different time delays over 950-2150 MHz. In terms of wavelength in a vacuum, this would be about 8 cm at 950 MHz, 5 cm at 1550 MHz and 3.5 cm at 2150 MHz. You can achieve these delays with coax cable, but the propagation velocity will be lower meaning these values should be reduced by about 25-30%. Conceptually if the two linear outputs for a band had a zero time offset, we would only have to insert a small additional amount of coax into one path before running them into a signal combiner (a splitter run the other way).
However the time delay offsets between the outputs is unknown, so one would have to experiment to find the proper constant offset. This isn't terribly hard if you can directly or indirectly measure CNR (sometimes called SQ). Just tweak for the best result for a specific frequency and calculate the appropriate length offsets for the other frequencies. I would guess you will need at the absolute minimum 5 different coax steps to cover the band, and preferably 10-20. And of course you will need to delay one way for left circular and the other for right.
You could do this coax selection manually, but that would be a pain. One method of partially automating this would be to run the fixed outputs from the LNBF into a 4x8 switch. Each receiver output would take two of the 4x8 outputs, one H and one V for the selected LNBF band. This would require a fairly simple circuit to pass through the polarization selection for one 4x8 output and reverse the polarity for the other. One output would go straight to a combiner, while the other would go to a 16x1 DiSEqC 1.1 switch with different selectable cable delays. You will have to play some games with diplexers and power inserters or diodes, inductors and capacitors to feed power and DiSEqC commands around the combiner. That part is left to the reader. Two inputs of a 4x1 DiSEqC switch could be used to select between linear and circular operation.
This is completely Rube Goldberg, but the actual cost is probably not that high - maybe $20-30 per receiver output, with four possible. Of course your receiver will require have to switch between the DiSEqC 1.1 outputs according to the transponder frequency. You could do this for even less if you want to mess around with cables. It makes much more sense to use a sidecar solution, but if you want to receiver circular signals over the entire universal range, that won't work. In North America, the ranges for circular signals does not cover more than 11.7-12.7 GHz to my knowledge.