Thermal conductivity of homogeneous twisted stacks of graphite is found to strongly depend on the misfit angle. The underlying mechanism relies on the angle dependence of phonon-phonon couplings across the twisted interface. Excellent agreement between the calculated thermal conductivity of narrow graphitic stacks and corresponding experimental results indicates the validity of the predictions. This is attributed to the accuracy of interlayer interactions descriptions obtained by the dedicated registry-dependent interlayer potential used. Similar results for h-BN stacks indicate overall higher conductivity and reduced misfit angle variation. This opens the way for the design of tunable heterogeneous junctions with controllable heat-transport properties ranging from substrate-isolation to efficient heat evacuation.