We present a systematic approach to the classification of the UV properties of higher dimensional operators spanned by $\Phi^\dagger \Phi$ (where $\Phi$ is the Higgs doublet) and ordinary derivatives thereof (e.g. $\partial_\mu(\Phi^\dagger \Phi) \partial^\mu(\Phi^\dagger \Phi), (\Phi^\dagger \Phi)^3, (\Phi^\dagger \Phi)^4 , \dots$) in Higgs Effective Field Theories. The procedure is purely algebraic and thus regularization-independent. It relies on a novel set of hidden symmetries that can be formulated in an extended field space where the singlet $\Phi^\dagger \Phi$ is treated as a dynamical variable. The resulting relations stemming from such symmetries are valid to all orders in the loop expansion for off-shell 1-PI Green's functions (and not only for $S$-matrix elements). One-loop applications are briefly discussed.