The superconducting diode effect manifests as nonreciprocal properties of critical currents, yet a systematic tensorial framework for its symmetry-governed responses remains lacking. We develop such a framework by relating the angular harmonics of critical currents to the magnetic field. Crystalline symmetries constrain the dipolar response to be purely antisymmetric (Hall-like) in Rashba superconductors with C₃ᵥ, C₄ᵥ, or C₆ᵥ symmetry, whereas they require a diagonal, traceless tensor in Dresselhaus superconductors with D_2d symmetry. For systems with fourfold symmetry, the octupole response appears at linear order, whose response tensors take symmetric off-diagonal and identity forms for C₄ᵥ and D_2d symmetries, respectively. Furthermore, nematicity activates otherwise forbidden tensor components, providing a sensitive probe of rotational symmetry breaking in superconductors. At cubic order in the magnetic field, the nonlinear response tensors likewise exhibit distinct symmetry-dependent patterns. Our framework establishes a symmetry-resolved characterization of the superconducting diode effect and provides a route to probing the underlying electronic structure and symmetry-breaking orders.