This is incorrect, the guarantee is that an orthogonal set of eigenvectors exist for symmetric matrices, but it is also possible to choose a non-orthogonal basis for subspaces corresponding to eigenvalues with multiplicity > 1. For example, the identity matrix in R^2 is symmetric, and all vectors in R^2 are its eigenvectors, we could choose any non-orthogonal pair to form the basis for the eigenvectors - if a given algorithm returned such an eigenvectors basis, this is perfectly correct, but we users may incorrectly assume basis is orthogonal.