Effective field theories (EFTs) suffer from a vast redundancy of description, reminiscent of coordinate invariance, that lends itself to a geometric treatment. In this talk I’ll survey recently-developed geometric insights into EFTs of the Standard Model Higgs sector, including invariant distinctions between possible EFTs of the Higgs boson, a new understanding of the connection between EFT geometry and observables, and generalizations of Riemannian field space geometry that encode information about analyticity and unitarity of the EFT. These developments are relevant to ongoing searches at the LHC and sharpen an open question for future colliders: is electroweak symmetry linearly realized by the known particles of the Standard Model?