We propose an alternative formulation of a left-right symmetric model where the difference between baryon number (B) and lepton number (L) remains an unbroken symmetry. This is unlike the conventional formulation, where B-L is promoted to a local symmetry and is broken explicitly in order to generate Majorana neutrino masses. In our case B-L remains a global symmetry after the left-right symmetry breaking, allowing only Dirac mass terms for neutrinos. In addition to parity restoration at some high scale, this formulation provides a natural framework to explain B-L as an anomaly-free global symmetry of the Standard Model and the nonobservation of (B-L)-violating processes. Neutrino masses are purely Dirac type and are generated either through a two-loop radiative mechanism or by implementing a Dirac seesaw mechanism.