How the non-metricity of the connection arises naturally in the classical theory of gravity

Three equivalent variational formulations of General Relativity Theory: 1) metric, 2) metric-affine (Palatini) and 3) purely affine, are discussed. The classical "Palatini method of variation" is thoroughly analysed. It implies non-metricity of the spacetime connection, as soon as the matter Lagrangian density is connection-sensitive. This conclusion is not new, but was never seriously discussed. We show, that accepting this non-metricity, i.e. treating both geometric structures of spacetime: the metric tensor and the connection, as being a priori independent, but interacting with each other via field equations, considerably simplifies the conceptual structure of the present Gravity Theory and, maybe, provides a better starting point for its generalisations from our Solar System scale to the Universe scale (dark matter).