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Two-dimensional Poincaré maps are widely used for discovering planar periodic and quasi-periodic orbits. These maps have well-defined and easily identifiable chains of islands that are indicative of quasi-periodic orbits. These chains of islands surround fixed points, which indicate periodic orbits. When expanding the Poincaré maps to four dimensions the problem becomes more complex, not only must the four dimensions be displayed in a manner that can be easily interpreted visually, but the structures within the Poincaré maps that indicate quasi-periodic motion are much more complex. An additional factor making these structures more complex is that spatial orbits can experience short-term, quasi-periodic behavior similar to that experienced by a planar orbit, but these spatial orbits can also undergo long-term, quasi-periodic behavior that causes periodic change in the shape of the orbit. In addition to providing indications of quasi-periodic behavior, the shape of the 4D Poincaré structures can provide insight into the transforming nature of the orbit. This paper uses the “space-plus-color” method for displaying the 4D Poincaré maps. Various structures within a 4D Poincaré map relating to spatial, quasi-periodic orbits in the Earth-Moon, circular restricted three-body problem are identified and analyzed. Trends among the structures are identified so that inferences can be made about the orbit relating to structures that have not yet been documented. Structures that clearly highlight a fixed point are identified, and a multiple-shooter method is used to converge on the periodic orbit. Similar structures are identified within 4D Poincaré maps with different Jacobi constants to demonstrate that these structures are not unique to a particular Jacobi constant. [ABSTRACT FROM AUTHOR]