We present experimental and theoretical studies of Pd/Cu(100) and Cu/Pd(100) heterostructures in order to explore their structure and misfit strain relaxation. Ultrathin Pd and Cu films are grown by pulsed laser deposition at room temperature. For Pd/Cu, compressive strain is released by networks of misfit dislocations running in the  and  directions, which appear after a few monolayers (ML) already. In striking contrast, for Cu/Pd the tensile overlayer remains coherent up to about 9 ML, after which multilayer growth occurs. The strong asymmetry between tensile and compressive cases is in contradiction with continuum elasticity theory and is also evident in the structural parameters of the strained films. Molecular dynamics calculations based on classical many-body potentials confirm the pronounced tensile-compressive asymmetry and are in good agreement with the experimental data.