In the Schwarzschild solution, the time orientation beyond the horizon at $r=2M$ can be found by considering a particle moving through it.
Since nothing physical is happening to the particle at $r=2M$ (as can be seen by using some other coordinate system, for example Penrose's), its future and past lightcones must point to the same spacetime directions before and after crossing. If the particle was entering the horizon from outside, his future lightcone pointed inwards just before entrance, and therefore must point inwards just after entering. This continuity argument shows that there is no white hole inside the horizon.
One may ask another question: whether there is another, different solution (not Schwarzschild) which looks like Schwarzchild outside the horizon, but is different inside. The answer to that is No - due to Birkhoff's theorem.