Let me write some notes on this.
Starlink's satellites do have a very distinct shape, largely one big and flat solar panel, mounted perpendicular on the further rather planar satellite body itself. And a who line of these satellites do line-up in the same orientation with their solar panel (of course slightly deviating according to their earth orbit). Be in the right spot, and you can see the sunlight mirroring in the solar panel, though not for all satellites at the same time.
When the whole (including the viewer) moves, the satellites giving the perfect line-up and providing the sunlight mirroring will "jump" among the line of satellites.
Another item relevant is the light polarization. Given the high-tech solar panel materials, the sunlight mirroring will depend on the polarization, or better, the opposite, where the mirroring will only let through specific polarization directions (which in turn also varies with the viewer position).
Compare this with the good old Polaroid sunglasses, taking away the sunlight reflection from the water surface in swimming pools. Polaroid sunglasses take away the horizontal polarized reflection glare according to the vertical polarization direction of the sunglasses. Turn your head 90 degrees flat, and you have the sunlight reflections in the water back.
Because the satellites as well as the viewer are moving in 3D, the light polarization at the viewer end is probably going to change, because the satellite mirroring surface is not a "mirror" though more a refraction pattern, with the effect that the mirroring is depending on the detailed surface structure of the solar panels as well the incident angles (more than one mirror angle) (so, it's not a bathroom mirror type of reflection).
Another aspect what may cause the perception of "dancing" satellites is, the anisotropic refraction characteristics of glass (cockpit windows). The result is that the same light source will start to end up as 2 or 3 points after it passes the glass. This even happens when the light goes perpendicular through the glass. This is a very strange aspect, not many people know about.
When, may moons ago, in university, my daily driver was a good old and rusty 1976 Austin Mini-van, where the rear door windows "doubled" the headlights of cars behind me. An offset of roughly 2–3 times the diameter of the headlight of the car behind me. The displacement angle was around 30 degrees with the vertical. When taught in physics college about the anisotropic refraction, it became clear to me, this was the reason of the double headlights visible in my car. In those days, there were no fancy led-lights, just ordinary filament bulbs, with no polarization, or whatever.
When such an anisotropic refraction happens, with light, where the polarization does change, the light source will seem to "move" between the 2 anisotropic refraction locations. I think, this is something happening with the described "light-show", where satellites seem to move around each-other (on top of the "next satellite mirroring). Looks very strange, until you understand what is happening with the polarization and the anisotropic refraction.
For those interested in this anisotropic refraction aspect, look-up the wiki articles around this subject.