Basically, split tones are produced through a slightly different embouchure position than normal. It is basically decentering the regular embouchure and placing greater tension in the middle, which forces the outside of the buzzing area to generate multiple resonance centers. The sounding result isn't a pure dyad, but is basically like a big vertical sonority based around this dyadic harmony (much like saxophone multiphonics). Conceptually, I view the split tone less as an even dyad, and more as a distortion of the upper note. I can "bend" the top note down to increase or decrease the amount of grit in the sound, but it's much more difficult to do this in the opposite direction without a kind of hitch or slot as I move up the harmonic series.
Interval size changes both the pitch and the timbre of these sounds, as demonstrated below.
Matt Barbier has also put together a really nice guide to these. It's geared towards learning and pedagogy, but might be useful reading!
The dynamic range is fairly limited - tending towards the louder side. Due to the relatively high amount of vibrating embouchure, I can't focus these down to soft dynamics like ordinario tone production. Certain registers I can really overdrive these techniques, and get a really startling sound - much louder than I can produce normally! These occur with top note roughly between G3 and Db4. Generally, the lower the split tone the lower its "resting" dynamic is, but once you start getting above Eb4, the strength of these sonorities starts to diminish.
In general, the best range is between Ab 2 and D4. I can go higher, up to Bb4, but since these intervals become so narrow, it gets harder to destabilize my conventional embouchure while keeping the alternative one engaged and stable enough to effectively hold the split tone. Provide high examples.
Regarding embouchure pressure (different from actually lip tension which helps to produce pitch) these split tones lie in the middle of a continuum from ordinario tone to granular sounds. In order to get into these splits, I can do one of two things. Bend the top pitch down, as if slurring very slowly, until I essentially bend into the middle of the two. The other is to over-focus my embouchure, increasing embouchure pressure until the sound contains fewer higher partials and sounds pinched. I can then pivot the horn a little bit and slide into the split tone without compromising the pitch of the upper note. But I can also just articulate the dyad immediately.
Once in the split tone territory, I can keep increasing embouchure pressure until the sound breaks even further into a more granulated and less resonant texture. More about this on its own page.
Essentially, I mean lip tension to be the speed or pitch in which my lips are buzzing, while embouchure pressure is the amount of pressure with which my lips are pressing against each other. This is almost identical to string overpressure and its signal-to-noise ratios.
Common tone splits
Since the actual execution of these split tones is more about pitch than it is about interval, you can easily slur between split tones that contain one pitch of the dyad in common (easiest on the upper note). We can work out specific instances, but generally there are these relationships on individual partials that are the same, or adjacent. One example would be a 4:3 split tone over the Bb fundamental (so, Bb3-F3) - once I engage the valve, converting the fundamental to F, I can execute another split tone, but this time 5:4 (A3-F3). Since these two dyads contain one pitch in common and I don't have to move the slide, I could slur between these two sonorities just by engaging the valve (even trilling!).
Pieces that extensively use split tones
Nicholas Deyoe - facesplitter (through harmon mute)
Clint McCallum - bowel resection (through solotone mute)
Iannis Xenakis - Keren (no mute)
Eric Wubbels - contraposition (through harmon mute)
Michelle lou - HoneyDripper (they're often hidden behind processing)
Andrew Greenwald - a thing 6 (on euphonium)