Any irreducible Dynkin diagram Delta is obtained from an irreducible Dynkin diagram Delta(h) of type ADE by folding via graph automorphisms. For any simple complex Lie group G with Dynkin diagram Delta and compact Riemann surface Sigma, we give a Lie-theoretic construction of families of quasi-projective Calabi-Yau threefolds together with an action of graph automorphisms over the Hitchin base associated to the pair (Sigma, G). They induce families of global quotient stacks with trivial canonical class, referred to as Calabi-Yau orbifolds, over the same base. Their intermediate Jacobian fibration, constructed in terms of equivariant cohomology, is isomorphic to the Hitchin system of the same type away from singular fibers. (C) 2018 Elsevier B.V. All rights reserved.