“I cannot believe God is a weak left-hander,”  quipped Pauli when it was discovered that weak interactions act only on left-handed particles.  However it is possible that at energies of ~10^14 or 10^15 GeV, which must have been present at the creation of the Universe, the laws of nature are invariant under parity or reflection, and do not discriminate between left and right handedness.  This is the likely energy scale for new physics indicated by the observed smallness of neutrino masses. But how can we test if nature is left-right symmetric at such high energy scales? In this paper we find that precision neutrino experiments that are looking for asymmetries in the interactions involving neutrinos and anti-neutrinos may be able to provide us the needed clue.  We make the testable prediction that these experiments will measure the  matter-anti matter symmetry (CP) violating phase to be consistent with zero (modulo \pi) in the leptonic sector.   The electric dipole moment of the neutron is known to be negligibly small. What our work shows is that in likely regions of parameter space of the simplest and  most elegant theory that restores left-right symmetry, an excessive neutron EDM is generated because of quantum effects from CP violating leptonic phases, and therefore these phases must be absent.  The work assumes that there are no axions, which are particles proposed by a popular competing theory to explain the smallness of neutron EDM.