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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. [ABSTRACT FROM AUTHOR]