Cilia and flagella are hairlike organelles that propel cells through fluid. The active motion of the axoneme, the motile structure inside cilia and flagella, is powered by molecular motors of the axonemal dynein family. These motors generate forces and torques that slide and bend the microtubule doublets within the axoneme. To create regular waveforms, the activities of the dyneins must be coordinated. It is thought that coordination is mediated by stresses due to radial, transverse, or sliding deformations, and which build up within the moving axoneme and feed back on dynein activity. However, which particular components of the stress regulate the motors to produce the observed waveforms of the many different types of flagella remains an open question. To address this question, we describe the axoneme as a three-dimensional bundle of filaments and characterize its mechanics. We show that regulation of the motors by radial and transverse stresses can lead to a coordinated flagellar motion only in the presence of twist. We show that twist, which could arise from torque produced by the dyneins, couples curvature to transverse and radial stresses. We calculate emergent beating patterns in twisted axonemes resulting from regulation by transverse stresses. The resulting waveforms are similar to those observed in flagella of Chlamydomonas and sperm. Due to the twist, the waveform has nonplanar components, which result in swimming trajectories such as twisted ribbons and helices, which agree with observations.