  High resolution VLBI observations revealed a quasi-stationary component (QSC)
in the relativistic jets of many blazars, which represents a standing
recollimation shock. The VLBA monitoring of the BL Lacertae jet at 15~GHz shows
the QSC at a projected distance of about 0.26~mas from the radio core. We study
the trajectory and kinematics of the QSC in BL Lacertae on sub-parsec scales
using 15~GHz VLBA maps of 164 observations over 20 years from the MOJAVE
program and 2~cm VLBA Survey. To reconstruct the QSC's intrinsic trajectory, we
use moving average and trajectory refinement procedures to smooth out the
effects of core displacement and account for QSC positioning errors. We
identified 22 QSC reversal patterns on spatial scales ranging from 0.01~mas to
0.05~mas and with a frequency of $\sim 1.5$ per year. Most reversals have an
acute angle $<90\degr$ at the turning point, and few have a loop-shaped or
arc-shaped trajectory. The directions of reversals (clockwise or
counterclockwise) appear to be random. Combinations of reversals show
reversible and quasi-periodic motion. We propose a model in which the reverse
motion of the QSC is due to the passage of a relativistic transverse jet wave
through a fixed location of the QSC, similar to the transverse motion of a
seagull on a wave. The QSC motion is governed by the amplitude, velocity, and
tilt of the relativistic transverse wave. According to the model, relativistic
waves are generated upstream of QSC. In the active state of the jet, the
directions of the twisting waves are random, similar to the behaviour of the
wave in a high-pressure hose, while in the jet stable state, the wave makes
quasi-periodic oscillations with regular twisting. In this model the transverse
speed of the QSC in the host galaxy frame is subluminal ($<0.3\,c$), but to an
observer it appears superluminal ($\sim 2\,c$).
