  We present a new scheme for teleporting a quantum state between two parties
whose local reference frames are misaligned by the action of a finite symmetry
group. Unlike other proposals, our scheme requires the same amount of classical
communication and entangled resources as conventional teleportation, does not
reveal any reference frame information, and is robust against changes in
reference frame alignment while the protocol is underway. The mathematical
foundation of our scheme is a unitary error basis which is permuted up to a
phase by the conjugation action of the group. We completely classify such
unitary error bases for qubits, exhibit constructions in higher dimension, and
provide a method for proving nonexistence in some cases.
