<p>In this paper, our focus lies on the study of the second-order variational analysis of orthogonally invariant matrix functions. It is well known that an orthogonally invariant matrix function is an extended-real-valued function that can be expressed as a composite function of absolutely symmetric functions and the singular values. We establish several second-order properties of orthogonally invariant matrix functions, such as parabolic epi-differentiability, parabolic regularity, and twice epi-differentiability when their associated absolutely symmetric functions enjoy some properties. Specifically, we show that the nuclear norm of a real matrix is twice epi-differentiable and we derive an explicit expression of its second-order epi-derivative. Moreover, for a convex orthogonally invariant matrix function, we calculate its second subderivative and present sufficient conditions for twice epi-differentiability. This enables us to establish second-order optimality conditions for a class of matrix optimization problems.</p>