  A real symmetric tensor is orthogonally decomposable (or odeco) if it can be
written as a linear combination of symmetric powers of $n$ vectors which form
an orthonormal basis of $\mathbb R^n$. Motivated by the spectral theorem for
real symmetric matrices, we study the properties of odeco tensors. We give a
formula for all of the eigenvectors of an odeco tensor. Moreover, we formulate
a set of polynomial equations that vanish on the odeco variety and we
conjecture that these polynomials generate its prime ideal. We prove this
conjecture in some cases and give strong evidence for its overall correctness.
