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We derive necessary and sufficient conditions for all global symmetries ofthe most general two Higgs doublet model (2HDM) scalar potential entirely interms of reparametrization independent, i.e. basis invariant, objects. Thisculminates in what we call a "Symmetry Map" of the parameter space of the modeland the fundamental insight that there are, in general, two algebraicallydistinct ways of how symmetries manifest themselves on basis invariant objects:either, basis invariant objects can be non-trivially related, or, basiscovariant objects can vanish. These two options have different consequences onthe resulting structure of the ring of basis invariants and on the number ofremaining physical parameters. Alongside, we derive for the first timenecessary and sufficient conditions for CP conservation in the 2HDM entirely interms of CP-even quantities. This study lays the methodological foundation foranalogous investigations of global symmetries in all other models that haveunphysical freedom of reparametrization, most notably the Standard Model flavorsector.

A fully basis invariant Symmetry Map of the 2HDM