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