I will discuss the distinction between linearly and non-linearly realized electroweak symmetry, or equivalently between the so-called SMEFT and HEFT families of effective field theories (EFTs). These theories can be classified according to the analyticity property of the Lagrangian as a function of the Higgs doublet H: the SMEFT Lagrangian is analytic, while that of the HEFT is non-analytic. This classification is not only simple and unambiguous, but also offers deeper physical insight. From the UV perspective, non-analyticity occurs when the new states acquire mass from electroweak symmetry breaking, and thus cannot be decoupled to arbitrarily high scales. This is reflected in the IR by the anomalous growth of the interaction strength for processes involving many Higgs bosons and longitudinally polarized massive vectors, with a breakdown of the EFT description at a scale $O(4 \pi v) \sim 3$ TeV. Conversely, analyticity occurs when new physics can be pushed parametrically above the electroweak scale. We illustrate the physical distinction between these two EFT families by discussing Higgs boson self-interactions.