Most web explanations of white holes seem to suggest that a hole is either black or white.
Not so.
They may be both at once.
Start with the Schwartzschield solution.
Familiar black hole features can be deduced from tracking in-going geodesics beyond the event horizon by switching to some other coordinate system that is smooth at the horizon.
Take the same hole and track the geodesic of an imaginary particle outside the hole traveling away fast enough to escape to infinity.
Track this geodesic backwards in time thru the event horizon.
This will require a different change of coordinates than those with which we tracked the in-falling particle, but there is only one manifold.
These two geodesics reach different singularities.
This is described in great detail in the Boyer, Lindquist paper “Maximal Analytic Extension of the Kerr Solution”.
See also page 834 of “Gravitation” by Misner, Thorne and Wheeler.
See this.
Kruskal-Szekeres coordinates are not singular at the event horizon.
See this.