Eigenvectors of a symmetric matrix and orthogonality

Given a full rank symmetric matrix $A_{p\times p}$ we can build a matrix $U=[u_1,...,u_p]$ where $u_i$ is the eigenvector associated to the $i^{th}$ largest eigenvalue of $A$.

Assuming that eigenvectors have unit norm it is easy to prove that $U'U=I_p$ (eigenvectors are orthogonal). I am wondering if somebody knows under which conditions it is also true that $UU'=I_p$