We consider a gauged $U{\left(1\right)}_{L_{\mu }-L_{\tau }}$ $$ \mathrm{U}{(1)}_{L_{\mu }-{L}_{\tau }} $$ 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 μτ of the $U{\left(1\right)}_{L_{\mu }-L_{\tau }}$ $$ \mathrm{U}{(1)}_{L_{\mu }-{L}_{\tau }} $$ symmetry with muons which can individually satisfy the current bounds on muon (g − 2) anomaly (∆a μ ). The other positive contributions to ∆a μ 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 ∆a μ 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 ∆a μ . 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.