For the past two decades, repeatable resonant tunneling transport of electrons in III-nitride double barrier heterostructures has remained elusive at room temperature. In this work we theoretically and experimentally study III-nitride double-barrier resonant tunneling diodes (RTDs), the quantum transport characteristics of which exhibit new features that are unexplainable using existing semiconductor theory. The repeatable and robust resonant transport in our devices enables us to track the origin of these features to the broken inversion symmetry in the uniaxial crystal structure, which generates built-in spontaneous and piezoelectric polarization fields. Resonant tunneling transport enabled by the ground state as well as by the first excited state is demonstrated for the first time over a wide temperature window in planar III-nitride RTDs. An analytical transport model for polar resonant tunneling heterostructures is introduced for the first time, showing a good quantitative agreement with experimental data. From this model we realize that tunneling transport is an extremely sensitive measure of the built-in polarization fields. Since such electric fields play a crucial role in the design of electronic and photonic devices, but are difficult to measure, our work provides a completely new method to accurately determine their magnitude for the entire class of polar heterostructures.