Real-order generalization of dissipativeness and passivity concepts are presented in this paper. They are characterized as properties of a system; that is, they are independent of the system's internal representation and independent of the type of fractional derivative defining that representation. With the aid of these extended concepts, the stability analysis of linearly interconnected multi-order (mixed-order or multivariable) linear or nonlinear systems consisting of integer and fractional order subsystems becomes a well-defined problem and it is reduced to verify algebraic inequalities and/or the dissipativenes of each subsystem. In particular, small gain and passivity theorems for multi-order systems are obtained. Examples show the benefits in simplicity obtained with this approach when analysing the stability of large-scale multi-order nonlinear systems.