Flexagon is an interesting two-dimensional geometric object that can be turned inside out through three-dimensional rotations. It is usually obtained by folding a long strip of paper into identical shapes such as equilateral triangles, squares, heptagons or hexagons and taping the two ends of the paper strip to form a closed loop. In this paper, a three-dimensional cuboid flexahedron is created based on its origami relative, tetraflexagon that consists of a loop of squares. The flexahedron is built by connecting identical cuboids (square prisms) through a special hinge-shifting mechanism. It demonstrates infinite turns whose faces can be hidden and revealed in a cyclic order through continuous flex moves. The kinematics of a tetraflexagon and a cuboid flexahedron are analysed and their cyclic motions are compared afterwards.