This paper develops a spectral framework for identification, estimation, and inference in non-Gaussian Structural Vector Autoregressive (SVAR) models using higher-order cumulants. Under independence or the absence of cross-cumulants, cumulant tensors of whitened innovations admit an orthogonal decomposition whose singular vectors recover the structural shocks. Identification is therefore governed by the spectral geometry of the population cumulant ten- sor. In particular, separation of tensor singular values provides a quantitative measure of identification strength through explicit perturbation bounds linking estimation error to the inverse singular-value gap. This characterization yields asymptotic normality under strong identification and nonstandard limits under local-to-weak identification sequences. We derive asymptotic distributions for tensor SVD estimators and show how statistically identified subsystems can be completed using conventional structural restrictions. Monte Carlo experiments and empirical applications illustrate the finite-sample properties and empirical relevance of the approach.