@Deltakosh here’s a Playground that shows what I’m trying to do.
It works, but it ain’t pretty. The only way I managed to get the desired result was by copying my skeleton and modifying it in steps.
Gray (srcSkeleton): Input data
Gray (rigSkeleton): This is the rig from Maya, it’s inside the purple skeleton so you can’t see it. The pose doesn’t match the input data so I can’t use it directly.
Purple (rigCopy): Copy of rigSkeleton. The way I copy it, I guess the transformation of the root bone is frozen.
Yellow (rotatedSkeleton): Rotated copy of rigCopy. This is aligned with the rest pose of the input data. I did not manage to just rotate an existing skeleton without messing something up so I came up with this method.
Green (targetSkeleton): Again, frozen version of rotatedSkeleton. This drives the mesh.
This all seems a bit crazy and it’s probably my lack of understanding how things work that led me to this complicated solution.
If anyone happen to come up with ways to simplify it I’m all ears 
UPDATE
This Playground shows the rest pose of the input data.
