should we be examining coordinateSystem to get the origin/direction into a common coordinate system before sending up to the browser process?"
probably?

At the moment eyeLevel is the "default" coordinate system, so anything given relative to that can be passed without transform.
(We'll probably need to change that when anchors come along, but fortunately that can be done invisibly to the user.)
for a stage frame of reference you'll need to multiply by the inverse of the sittingToStanding transform to get it back into the "default" coord sys.
and headModel is... weird.

I think we just multiply by the result of TransformBasePose, to get into "base coordinate system used by the service"

That transforms from the base coordinate system to the coordinate system in question. You want the inverse of that.
So you'll actually want to take something like TransformBaseInputPose (which handles headModel for input properly) and multiply do the inverse of whatever transform it eventually does.
https://cs.chromium.org/chromium/src/third_party/blink/renderer/modules/xr/xr_frame_of_reference.cc?sq=package:chromium&g=0&l=121

I think we also want to capture that matrix, then on the result side, transition the results back to the coordinate system passed into the request