  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 {\it
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 this theory has
an elegant spontaneous breaking of gauged scale symmetry and mass generation in
the absence of matter, where the necessary scalar field ($\phi$) is not added
ad-hoc to this purpose but is "extracted" from the $R^2$ term. The gauge field
becomes massive by absorbing the derivative term $\partial_\mu\ln\phi$ of the
Stueckelberg field ("dilaton"). In the broken phase one finds the
Einstein-Proca action of $v_\mu$ of mass proportional to the Planck scale
$M\sim \langle\phi\rangle$, and 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 scalar field (Higgs-like) with Palatini connection and
the potential is computed. In this case the theory 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 quadratic gravity of
similar symmetry, due to different non-metricity. This establishes a connection
between non-metricity and inflation predictions and enables us to test such
theories by future CMB experiments.
