  By exploiting a symmetric scheme for coupling $N$ spin-1/2 constituents (the
physical qubits) to states with total angular momentum $N/2-1$, we construct
rotationally invariant logical qudits of dimension $d=N-1$. One can encode all
qudit states, and realize all qudit measurements, by this construction. The
rotational invariance of all relevant objects enables one to transmit quantum
information without having aligned reference frames between the parties that
exchange the qudits. We illustrate the method by explicit constructions of
reference-frame-free qubits and qutrits and, for the qubit case, comment on
possible experimental implementations.
