  The lack of new physics at the LHC has sparked renewed interest in theories
of neutral naturalness, in which the Standard Model partners required for
naturalness carry no Standard Model charge. The Twin Higgs was the first of
these theories to be introduced, but recent work has demonstrated it is only an
isolated example in a large class of "orbifold Higgs" models. In this work we
study an orbifold Higgs model resulting from the orbifold projection by the
non-abelian group $S_3$. A model with multiple sectors uncharged under the SM
emerges. Constraints are placed on the model from Higgs phenomenology and the
prospects of finding evidence at colliders are discussed.
