Particle-antiparticle mixing, epsilon(K), Delta Gamma(q), A(SL)(q), A(CP)(B-d -> psi K-S), A(CP)(B-s -> psi phi) and B -> X-s,(d)gamma in the littlest higgs model with T-parity

We calculate a number of observables related to particle-antiparticle mixing in the Littlest Higgs model with T-parity (LHT). The resulting effective Hamiltonian for Delta F = 2 transitions agrees with the one of Hubisz et al., but our phenomenological analysis goes far beyond the one of these authors. In particular, we point out that the presence of mirror fermions with new flavour and CP-violating interactions allows to remove the possible Standard Model (SM) discrepancy between the CP asymmetry S psi(KS) and large values of vertical bar V-ub vertical bar and to obtain for the mass difference Delta Ms < (Delta M-s)(SM) as suggested by the recent result by the CDF collaboration. We also identify a scenario in which simultaneously significant enhancements of the CP asymmetries S-psi phi and A(SL)(q) relative to the SM are possible, while satisfying all existing constraints, in particular from the B -> X-s gamma decay and A(CP)(B -> X-s gamma) that are presented in the LHT model here for the first time. In another scenario the second, non-SM, value for the angle gamma = -(109 +/- 16)degrees from tree level decays, although unlikely, can be made consistent with all existing data with the help of mirror fermions. We present a number of correlations between the observables in question and study the implications of our results for the mass spectrum and the weak mixing matrix of mirror fermions. In the most interesting scenarios, the latter one turns out to have a hierarchical structure that differs significantly from the CKM one.