  We inspect the Littlest Higgs model with T-parity, based on a global symmetry
SU(5) spontaneously broken to SO(5), in order to elucidate the pathologies it
presents due to the non trivial interplay between the gauge invariance
associated to the heavy modes and the discrete T-parity symmetry. In
particular, the usual Yukawa Lagrangian responsible for providing masses to the
heavy `mirror' fermions is not gauge invariant. This is because it contains an
SO(5) quintuplet of right-handed fermions that transforms non-linearly under
SU(5), hence involving in general all SO(5) generators when a gauge
transformation is performed and not only those associated to its gauge
subgroup. Part of the solution to this problem consists of completing the
right-handed fermion quintuplet with T-odd `mirror partners' and a gauge
singlet, what has been previously suggested for other purposes. Furthermore, we
find that the singlet must be T-even, the global symmetry group must be
enlarged, an additional non-linear sigma field should be introduced to
parametrize the spontaneous symmetry breaking and new extra fermionic degrees
of freedom are required to give a mass to all fermions in an economic way while
preserving gauge invariance. Finally, we derive the Coleman-Weinberg potential
for the Goldstone fields using the background field method.
