Angles (lambda) at which parallel cortical microtubules (cMTs) were oriented with respect to the longitudinal direction were measured in Helianthus annuus hypocotyl epidermal cells. Histograms showing lambda frequencies in cell populations at the instant of epidermis fixation were obtained. Analysis of the histograms indicates that, in a particular position within a cell, the angle lambda changes periodically with time, i.e., there is a cycle of lambda change at that position. This cycle is most likely rotational rather than oscillatory, i.e., the change in lambda has a defined chirality (clockwise or counterclockwise). The full diversity of histograms can be consistently explained by rotational cycles with a variable velocity of lambda change, and with a cMT rebuilding stage taking place at a different phase of the cycle. The rotational cycles also provide the simplest explanation of cMT arrays in which the angle lambda changes along a cell (fixed) and no parallel orientation of cMTs is apparent at a certain position. This explanation assumes a gradient in the phase of the rotational cycle along the cell. The symmetry of the angular characteristics of the rotational cycle, with respect to the morphological directions in cells, leads to the concept that these directions typically represent the principal directions of a certain tensor quantity, which may control the cycling. Possible interactions between the rotational cycle of cMT reorientation and the helicoidal cycle during cell wall formation are discussed.