  A new systematic method for the explicit construction of (basis-)invariants
is introduced and employed to construct the full ring of basis invariants of
the Two-Higgs-Doublet-Model (2HDM) scalar sector. Co- and invariant quantities
are obtained by the use of hermitian projection operators. These projection
operators are constructed from Young tableaux via birdtrack diagrams and they
are used in two steps. First, to extract basis-covariant quantities, and
second, to combine the covariants in order to obtain the actual basis
invariants. The Hilbert series and Plethystic logarithm are used to find the
number and structure of the complete set of generating invariants as well as
their interrelations (syzygies). Having full control over the complete ring of
(CP-even and CP-odd) basis invariants, we give a new and simple proof of the
necessary and sufficient conditions for explicit CP conservation in the 2HDM,
confirming earlier results by Gunion and Haber. The method generalizes to other
models, with the only foreseeable limitation being computing power.
