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We provide a systematic study of minimal left-right models that are invariantunder $P$, $C$, and/or $CP$ transformations. Due to the high amount of symmetrysuch models are quite predictive in the amount and pattern of $CP$ violationthey can produce or accommodate at lower energies. Using current experimentalconstraints some of the models can already be excluded. For this purpose weprovide an overview of the experimental constraints on the different left-rightsymmetric models, considering bounds from colliders, meson-mixing andlow-energy observables, such as beta decay and electric dipole moments. Thefeatures of the various Yukawa and Higgs sectors are discussed in detail. Inparticular, we give the Higgs potentials for each case, discuss the possiblevacua and investigate the amount of fine-tuning present in these potentials. Itturns out that all left-right models with $P$, $C$, and/or $CP$ symmetry have ahigh degree of fine-tuning, unless supplemented with mechanisms to suppresscertain parameters. The models that are symmetric under both $P$ and $C$ arenot in accordance with present observations, whereas the models with either$P$, $C$, or $CP$ symmetry can not be excluded by data yet. To furtherconstrain and discriminate between the models measurements of $B$-mesonobservables at LHCb and $B$-factories will be especially important, whilemeasurements of the EDMs of light nuclei in particular could providecomplementary tests of the LRMs.

Viability of minimal left-right models with discrete symmetries