We construct the spin-projection operators for a theory containing a symmetric two-index tensor and a general three-index tensor. We then use them to analyze, at linearized level, the most general action for a metric affine theory of gravity with terms up to second order in curvature, which depends on 28 parameters. In the metric case, we recover known results. In the torsion-free case, we are able to determine the most general six-parameter class of theories that are projective invariant, contain only one massless spin 2 and no spin 3, and are free of ghosts and tachyons
