  We study the explicit form of Poincar\'e and discrete transformations of
flavor states in a two-flavor scalar model, which represents the simplest
example of the field mixing. Because of the particular form of the flavor
vacuum condensate, we find that the aforementioned symmetries are spontaneously
broken. The ensuing vacuum stability group is identified with the Euclidean
group E(3). With the help of Fabri-Picasso theorem, we show that flavor vacua
with different time labels and in different Lorentz frames are unitarily
inequivalent to each other and they constitute a manifold of zero-flavor-charge
states. Despite the spontaneous breakdown of Poincar\'e and CPT symmetries that
characterises such vacua, we provide arguments on the absence of Goldstone
Bosons. We also prove that the phenomenologically relevant oscillation formula
is invariant under these transformations.
