I am slowly conquering quaternions! Yesterday I got a conversion function from basis vectors to quaternion working (i.e. rotation matrix to quaternion) which is an important step to being able to answer some basic questions like "Give me a rotation where right face goes this way and up face goes that way."

Here's the weird little debug cube rendering I used while trying to get it working (Semi-transparent larger cube is the rotation if we convert from quaternion to basis vectors and then back again, it's now following the rotations of the main box. Yellow line is original quaternion axis and orange line is the semi-transparent box's quaternion axis)

I can share the implementation I landed on if anyone is interested but it mostly came from this page https://www.euclideanspace.com/maths/geometry/rotations/conversions/matrixToQuaternion/