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Electronic anisotropy (or `nematicity') has been detected in all main families of cuprate superconductors by a range of experimental techniques -- electronic Raman scattering, THz dichroism, thermal conductivity, torque magnetometry, second-harmonic generation -- and was directly visualized by scanning tunneling microscope (STM) spectroscopy. Using angle-resolved transverse resistance (ARTR) measurements, a very sensitive and background-free technique that can detect 0.5$\%$ anisotropy in transport, we have observed it also in La$_{2-x}$Sr$_{x}$CuO$_{4}$ (LSCO) for $0.02 \leq x \leq 0.25$. Arguably the key enigma in LSCO is the rotation of the nematic director with temperature; this has not been seen before in any material. Here, we address this puzzle by measuring the angle-resolved transverse magnetoresistance (ARTMR) in LSCO. We report a discovery of colossal transverse magnetoresistance (CTMR) -- an order-of-magnitude drop in the transverse resistivity in the magnetic field of $6\,$T, while none is seen in the longitudinal resistivity. We show that the apparent rotation of the nematic director is caused by superconducting phase fluctuations, which are much more anisotropic than the normal-electron fluid, and their respective directors are not parallel. This qualitative conclusion is robust and follows straight from the raw experimental data. We quantify this by modelling the measured (magneto-)conductivity by a sum of two conducting channels that correspond to distinct anisotropic Drude and Cooper-pair effective mass tensors. Strikingly, the anisotropy of Cooper-pair stiffness is significantly larger than that of the normal electrons, and it grows dramatically on the underdoped side, where the fluctuations become effectively quasi-one dimensional.