  We describe Wasserstein isometries of the quantum bit state space with
respect to distinguished cost operators. We derive a Wigner-type result for the
cost operator involving all the Pauli matrices: in this case, the isometry
group consists of unitary or anti-unitary conjugations. In the Bloch sphere
model, this means that the isometry group coincides with the classical symmetry
group $\mathbf{O}(3).$ On the other hand, for the cost generated by the qubit
"clock" and "shift" operators, we discovered non-surjective and non-injective
isometries as well, beyond the regular ones. This phenomenon mirrors certain
surprising properties of the quantum Wasserstein distance.
