"I am among other things using it for shape matching. I don't necessarily want to avoid a method based on a rest pose, I have already implemented this. It's just that I am almost completely sure it can be done without it. If a particle cloud's linear state, moment of inertia, angular momentum, velocity, and energy can be computed accurately from particle state, then so can the angle. I just haven't figured out how yet..."

I'm not an expert but I don't think that your conclusion necessarily follows.

It just doesn't make sense to me: particles only have a defined position, not an orientation. You can aggregate their positions to get a COM, which helps you define all the other properties you listed (eg angular momentum doesn't make sense for a particle in isolation, but does relative to the COM), but AFAICT the COM doesn't help define an orientation.

I'm struggling to see how you could define orientation without some sort of frame of reference similar to that provided by a rest pose.

This looks a bit like Wahba's problem:

https://en.wikipedia.org/wiki/Wahba%27s_problem

And you can e.g. use Davenport's method to solve it.

https://math.stackexchange.com/questions/1634113/davenports-q-method-finding-an-orientation-matching-a-set-of-point-samples

If you are looking to determine the inertia tensor of a rigid body object by summing up different masses/components, then there is a way. It is called the parallel axis theorem (I believe, It has been several years since I last looked into it) The idea is to find the inertia of a complex shape by breaking it down into simpler shapes with known properties (cubes, spheres, cones, etc) and then summing them up in a way analogous to summing up the different CoGs.

A good book that covers it is the O'Reilly Phyisics for Game developers, specifically the flight simulator section.

This looks a bit like Wahba's problem:

https://en.wikipedia.org/wiki/Wahba%27s_problem

And you can e.g. use Davenport's method to solve it.

https://math.stackexchange.com/questions/1634113/davenports-q-method-finding-an-orientation-matching-a-set-of-point-samples

AFAICT this still requires some idea of a "rest pose"/starting configuration (the a_i unit vectors in the stackexchange reply). (Thanks so much btw, I hadn't realized there were approaches to this besides Muller's! Very interesting.)

The OP wanted to know if there are any approaches that *don't* require some sort of initial state/rest pose.