Bean -- side address is like AKG 414. "End-fire" is like AKG 480s. Inverted would be like an upside-down AKG 414. (At least in terms of physical orientation of the microphone in space.)
Based on my limited knowledge, the beauty of soundfield mics - and the reason one may use any of those three orientations (side-address, end-address, inverted) - is that they capture the
entire sound field at a given point in space, i.e. 360° of sound in all three dimensions: vertical (up/down), lateral (side/side), depth (front/back). The reason this is possible: each capsule on a soundfield mic overlaps with the other capsules, ensuring total coverage in all directions (more on this in a moment). In post-processing, the user then may decide "how much" sound from each direction/dimension to use in the final mix. Kinda similar to M/S recording, where the user may control the width of the stereo image by deciding how much Side to mix in with the Mid signal. With a soundfield mic, the user doesn't just decide how much Side to mix in. The user decides how much sound to mix in
from each dimension: vertical (up/down), lateral (side/side), depth (front/back). In other words, the user may decide to "throw away" some (or all of) sound captured on the recording. For example, in a cruddy venue one may wish to throw away ALL the sound from the rear of the mic. (Think of how cardioids don't capture sound from behind them. Only in a soundfield mic's case, the sound is captured, but simply not used in the final mix.)
Another way to think of it: the mic captures an entire sphere of sound around the mic. Think of a single omnidirectional microphone that captures sound from all directions (for the sake of discussion, ignore that these nondirectional characteristics change to more directional characteristics at higher frequencies). With an omni mic, it's very simple (for the sake of discussion) - a single channel that includes sound from all directions. What you hear on that one channel is what you get - that's it. But with a soundfield mic, think of that entire sphere of music sliced into multiple, different 3-dimensional "wedges". The user may select which of those "wedges" to use, the extent to which one uses each, and which to discard outright.
The above is way overly simplistic, and not necessarily technically accurate in terms of how a soundfield mic works - it's just an attempt to help visualize the concept.
And Perhaps a graphic will help. Here's an image of a soundfield mic's pickup pattern. It includes 3 overlapping figure-8 patterns plus an omnidirectional pattern. If you can visualize how M/S may be decoded to control the stereo width, perhaps you can visualize how a soundfield mic allows the user to control not only the width (the figure-8 pattern oriented on the Y axis), but also the height (Z axis) and depth (X axis). (The omni mic is "W", which I've left out of the discussion for the time being.)
