We make a detailed analysis of the 13.5-day periodicity of the solar chromosphere, the near-Earth solar wind, interplanetary magnetic field and geomagnetic activity during the last three solar cycles. The 13.5-day periodicity is a real quasi-periodicity whose amplitude varies sizably with time, attaining occasionally values larger than, for example, the amplitude of the 27-day periodicity. In case of heliospheric and geomagnetic variables, intervals of large 13.5-day periodicity are due to the occurrence at 1 AU of two high-speed streams per solar rotation. According to the tilted solar dipole model, such two-stream structures appear if the heliospheric current sheet is sufficiently narrow and tilted. We show that, during the main two-stream structures, the interplanetary magnetic field (IMF) indeed had a persistent two-sector structure, and the heliosheet was sizably tilted. Multiple IMF sector structure is thus excluded as the main cause for 13.5-day periodicity in solar wind and geomagnetic activity. We determine the exact time and phase (solar longitude) of all intervals of significant 13.5-day periodicity during the last three solar cycles. We find that even the longest intervals of two-stream structure (up to 2 years) consist of separate activations. Each of the main activations of the 13.5-day (as well as 27-day) periodicity has a nearly equal length of a few (about 4) solar rotations only. This gives new, interesting information about the solar dynamics related to the development of the dipole tilt. Using the phase of the main 13.5-day activations, we could determine the longitudinal position of the solar dipole tilt for all major activations. We note that this position can abruptly change by even 90 deg between two successive 13.5-day activations. For each of the three solar cycles studied, the largest two-stream structures were found in the late declining phase of the cycle. On the other hand, the main activations of the 13.5-day periodicity of solar variables, which are due to two active solar longitudes approximately 180° apart, tend to occur around solar maxima.