Inelastic light scattering experiments on low-energy intrasubband spin-density excitations (SDEs) are performed in (001)-grown modulation-doped GaAs-AlGaAs single quantum wells in in-plane external magnetic fields. The investigated samples possess balanced linear Bychkov-Rashba (alpha) and Dresselhaus (beta) spin-orbit strengths in two different configurations, alpha = beta and alpha = -beta Both configurations lead to an extreme anisotropy of the spin splitting of the conduction band, where the in-plane directions of maximum spin splitting for both configurations are perpendicular to each other. The spin splitting asymmetry can be directly detected via the SDE by breaking of the time-reversal symmetry due to transfer of a momentum q in the quantum-well plane. In addition, the application of an in-plane magnetic field B-ext perpendicular to q allows us to modulate the effective magnetic field. Via a numerical line-shape analysis of the experimental SDE spectra, we determine the relevant parameters of the samples. We find that the linear spin-orbit strength vertical bar alpha vertical bar = beta is comparable for both samples, while the electron g factors are markedly different. Furthermore, we experimentally quantify the values of the maximum internal spin-orbit fields, which are as high as B-so similar to 18 T for both samples.