Hi Kyle, I'm having trouble following you.
Just to make sure we are all on the same page here let me try and clarify the diagrams above, both for myself and everyone else (Massive D, please correct me if I'm reading it wrong). As far as I can tell we are viewing a representation of two stereo pairs as viewed from above. The triangles represent individual cardioid microphones and the right angle point of each triangle indicates the direction in which each microphone is facing. The stage or sound source of interest is toward the top of the page. All good so far?
If we move the back pair forward until they are coincident with the front pair, we end up with two forward-facing coincident 90° X/Y positions spaced about 20cm apart (DIN spacing). If we swap all four cardioids for figure-8's we get.. well, it depends on the channel summing.. we get either 1) two Blumlein pairs near-spaced 20cm apart from each other. That's easy enough to imagine, except I don't expect a near spaced pair of them summed together would be useful. We'd have comb-filtering in both channels, from the sum of both right-facing fig-8's spaced 20cm apart into the right channel, and the sum of both left facing fig-8's into the left channel. ..or 2) We'd get a single pair of "virtual" figure-8's facing directly forward in a parallel 20cm spaced A-B arrangement. That's because if we sum the two figure-8's of a Blumlein pair the result is a single virtual fig-8 facing directly forward (the Mid channel of X/Y Blumlein converted to Mid/Side).
Okay enough of that brain-tease..
If you wanted to emulate a Blumlein pair using four cardioids instead of figure-8s, you'd place them all coincident and at 90° to each other, then swap the L/R routing and invert the polarity of the rear-facing two.
If instead of placing them coincident you were to place them outward facing in a near-spaced square ~20cm on each side, you form an IRT-Cross arrangement, which is a common recording format for 4-channel sound location ambiance beds. That works better for quad playback as its more open and spacious feeling than if you were to take the coincident four cardioid arrangement and fed each to its own speaker placed in opposing corners.
The inverse of that is fun to think about, and brings us back full circle to situations where a wide-spaced inward-angled arrangement may be applicable- imagine four cardioids placed at the vertices of a big square facing inward toward the center, with the sound sources located inside the square. Think festival camp jam under a square pop-up tent canopy with a small circle of performers playing under it. I always wanted to try hanging an ambisonic mic from the apex of such a canopy but never got around to it. This would be the inverse of that. In fact, the two techniques might actually work well in conjunction but would require 8 channels. Craziness!