  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.
