  While every matrix admits a singular value decomposition, in which the terms
are pairwise orthogonal in a strong sense, higher-order tensors typically do
not admit such an orthogonal decomposition. Those that do have attracted
attention from theoretical computer science and scientific computing. We
complement this existing body of literature with an algebro-geometric analysis
of the set of orthogonally decomposable tensors.
  More specifically, we prove that they form a real-algebraic variety defined
by polynomials of degree at most four. The exact degrees, and the corresponding
polynomials, are different in each of three times two scenarios: ordinary,
symmetric, or alternating tensors; and real-orthogonal versus complex-unitary.
A key feature of our approach is a surprising connection between orthogonally
decomposable tensors and semisimple algebras---associative in the ordinary and
symmetric settings and of compact Lie type in the alternating setting.
