Nowadays, the classical Internet has mainly envisioned as the underlying communication infrastructure of the Quantum Internet, aimed at providing services such as signaling and coordination messages. However, the interplay between classical and Quantum Internet is complex and its understanding is pivotal for an effective design of the Quantum Internet protocol stack. The aim of the paper is to shed the light on this interplay, by highlighting that such an interplay is indeed bidirectional rather than unidirectional. And the Quantum Internet exhibits the potential of supporting and even enhancing classical Internet functionalities.