Stable 2D solitons of semi-vortex (SV) and mixed-mode (MM) types are known to be maintained by the interaction of cubic attraction nonlinearities and spin-orbit coupling (SOC) in binary Bose-Einstein condensates. increase. We introduce a bilayer system in which two binary condensates stabilized by SOC are linearly coupled by tunneling. Numerical methods show that symmetric bilayer solitons undergo subcritical spontaneous symmetry breaking (SSB) bifurcations. The bifurcation produces a family of asymmetric 2D solitons. It exists up to a total norm value equal to the Townes soliton norm, above which collapse occurs. This situation terminates at a critical value of interlayer coupling, beyond which collapse begins prematurely and thus eliminates SSB bifurcations. Symmetric 2D solitons destabilized by SSB exhibit dynamic symmetry breaking in combination with the soliton’s intrinsic oscillations, or transitions to decay, if the soliton norm is large enough. Asymmetric MMs generated by SSB instability initiate spontaneous drifts in addition to intrinsic oscillations. Since SOC violates the Galilean invariance, considering 2D soliton transfer is an important issue. We found that the system supports moving the MM up to a critical value of velocity, beyond which it undergoes delocalization.