We will examine the muon g - 2 anomaly with the background of the Higgs global fit data in the framework of the Left-Right Twin Higgs (LRTH) Models. The joint constrains of the precision electroweak data, the 125 GeV Higgs data, the leptonic flavor changing decay mu -> e gamma decays, and the mass requirement of the right-handed neutrino nu(R), the vector-like top partner T and the heavy gauge boson W-H, m(nu R) > m(T) > m(WH), are all considered in our calculation. Furthermore, since the neutral scalar phi(0) may be lighter than the 125 GeV Higgs, the direct searches from the h -> phi(0)phi(0) channels can impose stringent upper limits on Br(h -> phi(0)phi(0)), which will reduce the allowed region of m(phi 0) and f, the vacuum expectation value of the SM right-handed Higgs H-R. It is concluded that the muon g-2 anomaly can be explained in the region of 700 GeV <= f <= 1100 GeV, 13 GeV <= m(phi 0) <= 55 GeV, 100 GeV <= m(phi +/-) <= 900 GeV, m(nu R) >= 15 TeV, and 200 GeV <= M <= 800 GeV, after imposing all the constraints mentioned above, where M here means the mass mixing coefficient M-(q) over bar LqR, allowed by gauge invariance.