We generalize the known equivalence between higher-order gravity theories and scalar–tensor theories to a new class of theories. Specifically, in the context of a first-order or Palatini variational principle where the metric and connection are treated as independent variables, we consider theories for which the Lagrangian density is a function f of (i) the Ricci scalar computed from the metric, and (ii) a second Ricci scalar computed from the connection. We show that such theories can be written as tensor–multi-scalar theories with two scalar fields with the following features: (i) the two-dimensional σ-model metric that defines the kinetic energy terms for the scalar fields has constant, negative curvature; (ii) the coupling function determining the coupling to matter of the scalar fields is universal, independent of the choice of function f; and (iii) if both mass eigenstates are long range, then the Eddington post-Newtonian parameter γ has value 1/2. Therefore, in order to be compatible with solar system experiments at least one of the mass eigenstates must be short range.