  We study the generalized version of energy-momentum squared gravity (EMSG) in
the Palatini formalism. This theory allows the existence of a scalar
constructed with energy-momentum tensor as $T_{\alpha\beta}T^{\alpha\beta}$ in
the generic action of the theory. We study the most general form of this theory
in the Palatini framework and present the underlying field equations. The
equations of motion of a massive test particle have been derived. The weak
field limit of the theory is explored and the generalized version of the
Poisson equation is obtained. Moreover, we explore the cosmological behavior of
the theory with emphasis on bouncing solutions. Some new bouncing solutions for
the specific Palatini EMSG model given by the Lagrangian density
$\mathcal{L}=R+\beta R^2+\eta T_{\mu\nu}T^{\mu\nu}$ are introduced. We show
that only the case $\eta>0$ can lead to viable cosmic bounce.
