Right, as a physicist, this may be super naive but matrices are just representations of the underlying algebraic structure so I'm not sure why it isn't possible to merely do both once you've decided on a way to convert between the two.
Rotors in geometric algebra form a “double cover” of rotations, for example using unit quaternions there are two different quaternions (q and -q) which will map to the same rotation matrix. This means that mapping from rotation matrices to quaternions is a bit fraught - you need to pick a “side”, and if that side happens to be opposite to what another part of the code (perhaps the part only using quaternions) chose, you’ll get some whacky stuff happening. The same idea applies for geometric algebra.
