Tilted-component systems are known to be characterized by aberrations with unusual field dependences, such as decentered coma and binodal astigmatism. In this paper, the origin of binodal astigmatism in a multielement system from the contributions of individual surfaces is explained in an intuitive manner, as a logical extension of the ordinary aberrations known to all optical designers. The insight provided by this graphical model allows an understanding of why the astigmatism of any given system behaves the way it does, and what needs to be done to convert it to ordinary quadratic behavior, so that what remains can be corrected by a final, rotationally symmetric subsystem. Conditions under which ordinary behavior may be obtained from a system of tilted elements are explored. It is shown that a certain type of symmetry of the primary aberrations may be obtained (regardless of the tilt angles) if the initial, untilted system is monocentric. Graphical displays of the aberration types are used to explore the behavior of the system across the field, and direct optimization of the Zernike coefficients is used to guide the performance of the system.