The gravitational form factors for a hadron, the form factors for the hadron matrix element of the QCD energy-momentum tensor, not only describe the coupling of the hadron with a graviton, but also serve as unique quantities for describing the shape inside the hadron reflecting dynamics of quarks and gluons, such as the internal shear forces acting on the quarks/gluons and their pressure distributions. The gravitational form factors are also relevant to understanding the origin of the nucleon mass and the origin of the nucleon spin, which are the important objectives at J-PARC. We consider the quark/gluon contributions to the gravitational form factors for a hadron, in particular, for a (pseudo)scalar hadron and for the nucleon. We derive and clarify the relations satisfied by the gravitational form factors as direct consequences of the symmetries and the equations of motion in QCD, and connections to the generalized parton distributions (GPDs). Our results reveal the connections between the gravitational form factors and the higher-twist quark-gluon correlation effects inside the hadrons. Furthermore, we are able to constrain the twist-four gravitational form factors by the trace anomalies in QCD, and derive the corresponding relations at the three-loop level.