  Electroweak precision tests put stringent constraints on the parameter space
of little Higgs models. Tree-level exchange of TeV scale particles in a generic
little Higgs model produce higher dimensional operators that make contributions
to electroweak observables that are typically too large. To avoid this problem
a discrete symmetry dubbed T-parity can be introduced to forbid the dangerous
couplings. However, it was realized that in simple group models such as the
littlest Higgs model, the implementation of T-parity in a UV completion could
present some challenges. The situation is analogous to the one in QCD where the
pion can easily be defined as being odd under a new $Z_2$ symmetry in the
chiral Lagrangian, but this $Z_2$ is not a symmetry of the quark Lagrangian. In
this paper we examine the possibility of implementing a T-parity in the low
energy $SU(6)/Sp(6)$ model that might be easier to realize in the UV. In our
model, the T-parity acts on the low energy non-linear sigma model field in way
which is different to what was originally proposed for the Littlest Higgs, and
lead to a different low energy theory. In particular, the Higgs sector of this
model is a inert two Higgs doublets model with an approximate custodial
symmetry. We examine the contributions of the various sectors of the model to
electroweak precision data, and to the dark matter abundance.
