Based on the mechanism of four-fold rigid origami, this study proposes a type of deployable truss structures that consist of repetitive basic parts and retain full cyclic symmetry in the folding/deployment process. On the basis of the irreducible representations and the great orthogonality theorem, symmetry-adapted analysis using group theory is described to identify the symmetry of mobility and kinematic behavior. Equivalent three-dimensional pin-jointed frameworks are employed for the symmetric structures. To verify that the structures can be foldable while retaining their full symmetries, numerical simulations on a series of structures with different symmetries and geometries are carried out. An artificial damping is introduced to stabilize the nonlinear folding behavior with singularity. Symmetry-adapted mobility analysis reveals that the structures of this type can be continuously folded with one degree-of-freedoms. Numerical simulations using the nonlinear iterative method accurately predict the folding behavior, as the results agree very well with the theoretic value.