We consider a gauged $U(1)_{L_mu-L_tau}$ extension of the left-right symmetric theory in order to simultaneously explain neutrino mass, mixing and the muon anomalous magnetic moment. We get sizeable contribution from the interaction of the new light gauge boson $Z_{mutau}$ of the $U(1)_{L_mu-L_tau}$ symmetry with muons which can individually satisfy the current bounds on muon $(g-2)$ anomaly ($Delta a_mu$). The other positive contributions to $Delta a_mu$ come from the interactions of singly charged gauge bosons $W_L$, $W_R$ with heavy neutral fermions and that of neutral CP-even scalars with muons. The interaction of $W_L$ with heavy neutrino is facilitated by inverse seesaw mechanism which allows large light-heavy neutrino mixing and explains neutrino mass in our model. CP-even scalars with mass around few hundreds GeV can also satisfy the entire current muon anomaly bound. The results show that the model gives a small but non-negligible contribution to $Delta a_mu$ thereby eliminating the entire deviation in theoretical prediction and experimental result of muon $(g-2)$ anomaly. We have briefly presented a comparative study for symmetric and asymmetric left-right symmetric model in context of various contribution to $Delta a_mu$. We also discuss how the generation of neutrino mass is affected when left-right symmetry breaks down to Standard Model symmetry via various choices of scalars.
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