  In this paper, we study the Gelfand--Cetlin systems and polytopes of the
co-adjoint $\mathrm{SO}(n)$-orbits. We describe the face structure of
Gelfand--Cetlin polytopes and iterated bundle structure of Gelfand--Cetlin
fibers in terms of combinatorics on the ladder diagrams. Using this
description, we classify all Lagrangian fibers.
