  This is the first paper in a series devoted to understanding the classical
and quantum nature of edge modes and symmetries in gravitational systems. The
goal of this analysis is to: i) achieve a clear understanding of how different
formulations of gravity provide non-trivial representations of different
sectors of the corner symmetry algebra, and ii) set the foundations of a new
proposal for states of quantum geometry as representation states of this corner
symmetry algebra. In this first paper we explain how different formulations of
gravity, in both metric and tetrad variables, share the same bulk symplectic
structure but differ at the corner, and in turn lead to inequivalent
representations of the corner symmetry algebra. This provides an organizing
criterion for formulations of gravity depending on how big the physical
symmetry group that is non-trivially represented at the corner is. This
principle can be used as a "treasure map" revealing new clues and routes in the
quest for quantum gravity. Building up on these results, we perform a detailed
analysis of the corner symplectic potential and symmetries of
Einstein-Cartan-Holst gravity in [1], use this to provide a new look at the
simplicity constraints in [2], and tackle the quantization in [3].
