This paper presents a novel discovery of a symmetry-breaking effect in porous media with porosity between 0.8-0.9, which we are referring to as the intermediate porosity flow regime. Using large eddy simulation, we studied how heat transfer and turbulent convection occurs within these materials at a microscopic level. We observed symmetry-breaking in porous structures made of regularly spaced circular cylinders, a common design in heat exchangers, immediately following the laminar to turbulent flow transition between Reynolds numbers of 37 and 100. Asymmetric patterns persisted up to Reynolds numbers of 1,000. The initial breakdown of symmetry occurs through a Hopf bifurcation, creating an oscillating flow pattern as shear layers interact around the solid obstacles. When the flow becomes turbulent, random variations in the timing of vortex oscillations (caused by the secondary instability) create asymmetric distributions of fluid velocity and temperature throughout the porous space. This leads to the formation of alternating channels with high and low velocity fluid flow. At the macroscale level, this loss of symmetry creates residual transverse drag force components and asymmetric heat flux distribution on the solid obstacle surfaces. Interestingly, the oscillating flow pattern promotes attached flow on the circular cylinder surfaces, which enhances heat transfer from the cylinders to the fluid. We observe that this secondary flow instability is the primary mechanism of enhanced turbulent heat flux from porous media with circular cylinders compared to those with square cylinders.