In general relativity, the only dynamical field describing the gravitational interaction of matter, is the metric. It induces the causal structure of spacetime, governs the motion of physical bodies through its Levi-Civita connection, and mediates gravity via the curvature of this connection. While numerous modified theories of gravity retain these principles, it is also possible to introduce another affine connection as a fundamental field, and consider its properties - curvature, torsion, nonmetricity - as the m…