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Using recent results, I will discuss what insight can be gained in general relativity by taking seriously Klein's Erlangen Program and Cartan's extension of it to differential geometry. The talk will be divided into three main parts. First, I will review how Cartan geometry gives us the geometry of spacetime by "breaking" SO(4,1) symmetry to SO(3,1), as in the MacDowell-Mansouri-Stelle-West formulation of gravity. Second, I will describe how a "field of observers" breaks the symmetry further to SO(3), yielding a system of evolving spatial Cartan geometries, or "Cartan geometrodynamics". This gives a manifestly Lorentz-covariant counterpart of the Ashtekar-Barbero formulation, which is the classical starting point for loop quantum gravity. Third, I will combine the first two parts, and consider breaking SO(4,1) to SO(3) directly. This describes the geometry of "observer space", the bundle of future-timelike unit vectors in spacetime, giving a perspective with key features of both the spacetime and geometrodynamic approaches. I will also discuss how observer space geometry provides a unified geometric setting for relating various alternative theories of gravity.