The appearance of wavelike helical structures on steady relativistic jets is studied using a normal mode analysis of the linearized fluid equations. Helical structures produced by the normal modes scale relative to the resonant (most unstable) wavelength and not with the absolute wavelength. The resonant wavelength of the normal modes can be less than the jet radius even on highly relativistic jets. High-pressure regions helically twisted around the jet beam may be confined close to the jet surface, penetrate deeply into the jet interior, or be confined to the jet interior. The high-pressure regions range from thin and ribbon-like to thick and tubelike depending on the mode and wavelength. The wave speeds can be significantly different at different wavelengths but are less than the flow speed. The highest wave speed for the jets studied has a Lorentz factor somewhat more than half that of the underlying flow speed. A maximum pressure fluctuation criterion found through comparison between theory and a set of relativistic axisymmetric jet simulations is applied to estimate the maximum amplitudes of the helical, elliptical, and triangular normal modes. Transverse velocity fluctuations for these asymmetric modes are up to twice the amplitude of those associated with the axisymmetric pinch mode. The maximum amplitude of jet distortions and the accompanying velocity fluctuations at, for example, the resonant wavelength decreases as the Lorentz factor increases. Long-wavelength helical surface mode and shorter wavelength helical first body mode generated structures should be the most significant. Emission from high-pressure regions as they twist around the jet beam can vary significantly as a result of angular variation in the flow direction associated with normal mode structures if they are viewed at about the beaming angle theta = 1/gamma. Variation in the Doppler boost factor can lead to brightness asymmetries by factors up to 6 as long-wavelength helical structure produced by the helical surface mode winds around the jet. Higher order surface modes and first body modes produce less variation. Angular variation in the flow direction associated with the helical mode appears consistent with precessing jet models that have been proposed to explain the variability in 3C 273 and BL Lac object AO 0235+164. In particular, cyclic angular variation in the flow direction produced by the normal modes could produce the activity seen in BL Lac object OJ 287. Jet precession provides a mechanism for triggering the helical modes on multiple length scales, e.g., the galactic superluminal GRO J1655-40.
