The reciprocity of dissipative systems is usually justified by the microscopic reversibility of physical processes, i.e., relying on the time-reversal symmetry of physical laws at the microscopic level. Here, it is shown that it is unnecessary to invoke microscopic arguments to establish a direct link between the reciprocity of macroscopic systems and time-reversal invariance. It is demonstrated that lossy dielectrics have a hidden time-reversal symmetry, as the relevant dissipation channels can be mimicked by a distributed network of time-reversal invariant lossless transmission lines. It is proven that the reciprocity of lossy systems is fundamentally rooted on the hidden time-reversal invariance and linearity of the materials. Furthermore, it is demonstrated that the upper-half frequency plane response of dissipative materials can be approximated as much as desired by the response of some lossless material. The developed theory sheds new light on the link between dissipation, "open systems," and interactions with a "bath" of oscillators.