Directional Andreev-Reflection Signatures of Inter-Orbital Pairing in Sr$_2$RuO$_4$

Unconventional superconductivity in quasi--two-dimensional systems is commonly identified through the emergence of Andreev bound states (ABS) at in-plane edges, while surfaces perpendicular to out-of-plane direction remain fully gapped due to weak interlayer coherence. This directional anisotropy has long served as a key paradigm for constraining pairing symmetries. Here, we show that Sr$_2$RuO$_4$ exhibits a striking reversal of this behavior. Using edge- and surface-sensitive spectroscopy, we observe pronounced in-gap ABS at surfaces perpendicular to the out-of-plane direction, whereas in-plane edges exhibit a reduced intensity of the in-gap spectral features. We show that this anomalous anisotropy can arise from the inter-orbital character of the superconducting pairing. Both even- and odd-parity inter-orbital pairing channels naturally generate robust surface ABS while suppressing planar edge modes and can also provide a mechanism for the appearance of a horizontal line node. Supported by \textit{ab initio} and model calculations, including Sr$_2$RuO$_4$/Ag interface reconstructions, our results highlight the possible role of inter-orbital correlations in shaping the spectroscopic response and provide constraints on the structure of the superconducting order parameter in Sr$_2$RuO$_4$.