In this paper, we analyze the polarized muon decay at rest admitting the non-standard \(V+A\) interaction in addition to the standard \(V-A\) interaction. We show that the angle–energy distribution of Dirac muon neutrinos (\(\nu _{\mu }\)’) is sensitive to the interference terms between the standard \(V-A\) coupling of left-chirality \(\nu _{\mu }\) and exotic \(V+A\) coupling of right-chirality one, which are proportional to the transverse components of \(\nu _{\mu }\) spin polarization. The interferences do not vanish in the limit of massless \(\nu _{\mu }\) and include the relative phases to test the CP violation. It allows one to calculate a neutrino flux for an assumed configuration of detector in the case of standard model prediction, and the upper limits on the neutrino flux for the left–right \(\nu _{\mu }\) mixture. It is also demonstrated that the effects of neutrino mass and mixing are tiny corrections and can be omitted. The most important conclusion is that even in the massless \(\nu _{\mu }\) limit, there are physical effects connected with the mixture of left- and right-handed helicity components in a spin \(1/2\) quantum state, when the exotic interaction \(V+A\) is admitted, contrary to the electron observables, where all the possible interferences vanish.