  The Palatini formalism is developed for gravitational theories in flat
geometries. We focus on two particularly interesting scenarios. First, we fix
the connection to be metric compatible, but we follow a completely covariant
approach by imposing the constraints with suitable Lagrange multipliers. For a
general quadratic theory we show how torsion naturally propagates and we
reproduce the Teleparallel Equivalent of General Relativity as a particular
quadratic action that features an additional Lorentz symmetry. We then study
the much less explored theories formulated in a geometry with neither curvature
nor torsion, so that all the geometrical information is encoded in the
non-metricity. We discuss how this geometrical framework leads to a purely
inertial connection that can thus be completely removed by a coordinate gauge
choice, the coincident gauge. From the quadratic theory we recover a simpler
formulation of General Relativity in the form of the Einstein action, which
enjoys an enhanced symmetry that reduces to a second linearised diffeomorphism
at linear order. More general theories in both geometries can be formulated
consistently by taking into account the inertial connection and the associated
additional degrees of freedom. As immediate applications, the new cosmological
equations and their Newtonian limit are considered, where the role of the lapse
in the consistency of the equations is clarified, and the Schwarzschild black
hole entropy is computed by evaluating the corresponding Euclidean action. We
discuss how the boundary terms in the usual formulation of General Relativity
are related to different choices of coordinates in its coincident version and
show that in isotropic coordinates the Euclidean action is finite without the
need to introduce boundary or normalisation terms.
