  The transition from classical to quantum mechanics has intrigued scientists
in the past and remains one of the most fundamental conceptual challenges in
state-of-the-art physics. Beyond the quantum mechanical correspondence
principle, quantum-classical analogies have attracted considerable interest. In
this work, we present classical two-mode interference for a nanomechanical
two-mode system, realizing classical St\"uckelberg interferometry. In the past,
St\"uckelberg interferometry has been investigated exclusively in quantum
mechanical two-level systems. Here, we experimentally demonstrate a classical
analog of St\"uckelberg interferometry taking advantage of coherent energy
exchange between two-strongly coupled, high quality factor nanomechanical
resonator modes. Furthermore, we provide an exact theoretical solution for the
double passage St\"uckelberg problem which reveals the analogy of the return
probabilities in the quantum mechanical and the classical version of the
problem. This result qualifies classical two-mode systems at large as a testbed
for quantum mechanical interferometry.
