  We develop quantum information processing primitives for the planar rotor,
the state space of a particle on a circle. By interpreting rotor wavefunctions
as periodically identified wavefunctions of a harmonic oscillator, we determine
the group of bosonic Gaussian operations inherited by the rotor. This $n$-rotor
Clifford group, $\text{U}(1)^{n(n+1)/2} \rtimes \text{GL}_n(\mathbb{Z})$, is
represented by continuous $\text{U}(1)$ gates generated by polynomials
quadratic in angular momenta, as well as discrete $\text{GL}_n(\mathbb Z)$
momentum sign-flip and sum gates. We classify homological rotor
error-correcting codes [arXiv:2303.13723] and various rotor states based on
equivalence under Clifford operations.
  Reversing direction, we map homological rotor codes and rotor Clifford
operations back into oscillators by interpreting occupation-number states as
rotor states of non-negative angular momentum. This yields new multimode
homological bosonic codes protecting against dephasing and changes in
occupation number, along with their corresponding encoding and decoding
circuits. In particular, we show how to non-destructively measure the
oscillator phase using conditional occupation-number addition and post
selection. We also outline several rotor and oscillator varieties of the
GKP-stabilizer codes [arXiv:1903.12615].
