  Surface bundles arising from periodic mapping classes may sometimes have
non-isomorphic, but profinitely isomorphic fundamental groups. Pairs of this
kind have been discovered by Hempel. This paper exhibits examples of nontrivial
Hempel pairs where the mapping tori can be distinguished by some Turaev--Viro
invariants, and also examples where they cannot be distinguished by any
Turaev--Viro invariants.
