  We formally define homological quantum rotor codes which use multiple quantum
rotors to encode logical information. These codes generalize homological or CSS
quantum codes for qubits or qudits, as well as linear oscillator codes which
encode logical oscillators. Unlike for qubits or oscillators, homological
quantum rotor codes allow one to encode both logical rotors and logical qudits
in the same block of code, depending on the homology of the underlying chain
complex. In particular, a code based on the chain complex obtained from
tessellating the real projective plane or a M\"{o}bius strip encodes a qubit.
We discuss the distance scaling for such codes which can be more subtle than in
the qubit case due to the concept of logical operator spreading by continuous
stabilizer phase-shifts. We give constructions of homological quantum rotor
codes based on 2D and 3D manifolds as well as products of chain complexes.
Superconducting devices being composed of islands with integer Cooper pair
charges could form a natural hardware platform for realizing these codes: we
show that the $0$-$\pi$-qubit as well as Kitaev's current-mirror qubit -- also
known as the M\"{o}bius strip qubit -- are indeed small examples of such codes
and discuss possible extensions.
