  A first-order gauge invariant formulation for the two-dimensional quantum
rigid rotor is long known in the theoretical physics community as an isolated
peculiar model. Parallel to that fact, the longstanding constraints
abelianization problem, aiming at the conversion from second to first class
systems for quantization purposes, has been approached a number of times in the
literature with a handful of different forms and techniques and still continues
to be a source of lively and interesting discussions. Connecting these two
points, we develop a new systematic method for converting second class systems
to first class ones, valid for a class of systems encompassing the quantum
rigid rotor as a special case. In particular the gauge invariance of the
quantum rigid rotor is fully clarified and generalized in the context of
arbitrary translations along the radial momentum direction. Our method differs
substantially from previous ones as it does not rely neither on the
introduction of new auxiliary variables nor on the a priori interpretation of
the second class constraints as coming from a gauge-fixing process.
