In this work, we theoretically construct exact mappings of many-particle bosonic systems onto quantum rotor models. In particular, we analyze the rotor representation of spinor Bose-Einstein condensates. In a previous work [R. Barnett et al., Phys. Rev. A 82, 031602(R) (2010)] it was shown that there is an exact mapping of a spin-one condensate of fixed particle number with quadratic Zeeman interaction onto a quantum rotor model. Since the rotor model has an unbounded spectrum from above, it has many more eigenstates than the original bosonic model. Here we show that for each subset of states with fixed spin F-z, the physical rotor eigenstates are always those with the lowest energy. We classify three distinct physical limits of the rotor model: the Rabi, Josephson, and Fock regimes. The last regime corresponds to a fragmented condensate and is thus not captured by the Bogoliubov theory. We next consider the semiclassical limit of the rotor problem and make connections with the quantum wave functions through the use of the Husimi distribution function. Finally, we describe how to extend the analysis to higher-spin systems and derive a rotor model for the spin-two condensate. Theoretical details of the rotor mapping are also provided here.