  Bosonic rotation codes, introduced here, are a broad class of bosonic
error-correcting codes based on phase-space rotation symmetry. We present a
universal quantum computing scheme applicable to a subset of this
class--number-phase codes--which includes the well-known cat and binomial
codes, among many others. The entangling gate in our scheme is code-agnostic
and can be used to interface different rotation-symmetric encodings. In
addition to a universal set of operations, we propose a teleportation-based
error correction scheme that allows recoveries to be tracked entirely in
software. Focusing on cat and binomial codes as examples, we compute average
gate fidelities for error correction under simultaneous loss and dephasing
noise and show numerically that the error-correction scheme is close to optimal
for error-free ancillae and ideal measurements. Finally, we present a scheme
for fault-tolerant, universal quantum computing based on concatenation of
number-phase codes and Bacon-Shor subsystem codes.
