We study quadratic gravity R ^ { 2 } + R _ { [ \mu \nu ] } ^ { 2 } in the Palatini formalism where the connection and the metric are independent .
This action has a gauged scale symmetry ( also known as Weyl gauge symmetry ) of Weyl gauge field v _ { \mu } = ( \tilde { \Gamma } _ { \mu } - \Gamma _ { \mu } ) / 2 , with \tilde { \Gamma } _ { \mu } ( \Gamma _ { \mu } ) the trace of the Palatini ( Levi-Civita ) connection , respectively .
The underlying geometry is non-metric due to the R _ { [ \mu \nu ] } ^ { 2 } term acting as a gauge kinetic term for v _ { \mu } .
We show that the gauge field becomes massive by a gravitational Stueckelberg mechanism by absorbing the derivative of the dilaton ( \partial _ { \mu } \ln \phi ) .
Palatini quadratic gravity with dynamical \tilde { \Gamma } _ { \mu } \sim v _ { \mu } is thus a gauged scale invariant theory broken spontaneously .
In the broken phase one finds the Einstein-Proca action of v _ { \mu } of mass near the Planck scale ( M ) with a positive cosmological constant .
Below this scale v _ { \mu } decouples , the connection becomes Levi-Civita and metricity and Einstein gravity are recovered .
These results remain valid in the presence of non-minimally coupled matter , with Palatini connection .
This is similar to recent results by the author for Weyl quadratic gravity , up to different non-metricity effects .
When coupled to a Higgs-like scalar field , Palatini quadratic gravity gives successful inflation and a specific prediction for the tensor-to-scalar ratio 0.007 \leq r \leq 0.01 for current spectral index n _ { s } ( at 95 \% CL ) and N = 60 efolds .
This value of r is mildly larger than in inflation in Weyl gravity , due to different non-metricity .
This establishes a connection between non-metricity and inflation predictions and enables us to test these theories by future CMB experiments .