We present a TeV-scale left-right ultraviolet completion of type-I seesaw for neutrino masses based on the $SU(2)_L\times SU(2)_R\times U(1)_{B-L}$ gauge group without parity, which leads to "large" light-heavy neutrino mixing while keeping the neutrino masses small in a natural manner guaranteed by discrete symmetries. We point out specific observable implications of this class of models if the $SU(2)_R$-breaking scale is of order 5 TeV, in searches for lepton flavor violating processes such as $μ\to eγ$, $μ\to 3 e$ and $μ-e$ conversion in nuclei, and lepton number violating processes such as neutrinoless double beta decay as well as at the LHC. In particular, if the upper limit on BR$(μ\to eγ)$ improves by one order of magnitude, a large range of the parameters of the model would be ruled out. ; Comment: 34 pages, 8 figures, 10 tables; some comments and references added; version accepted for publication in Phys. Rev. D