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In the III region of the maximally extended Schwarzschild solution, described in the Kruskal coordinates , Sch. time runs in the reversed direction as compared to the I - event horizon III-II of the photon from III is at the past infinity, while the III-IV is at the future infinity. This is refered to as a ``mirror wormhole''

I have seen that this is sometimes interpreted as if the both I and III have the same time direction? How so?

Does this means that the future singularity of the photon emitted in I is the past singularity of the photon emitted in III? Vice versa, is the past singularity of the photon from I future singularity of the photon in III. In other words, do Black Hole and White Hole exchange position between I and III?

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In the III region of the maximally extended Schwarzschild solution, described in the Kruskal coordinates , Sch. time runs in the reversed direction as compared to the I - event horizon III-II of the photon from III is at the past infinity, while the III-IV is at the future infinity.

Not true. This is an obscure attempted reinterpretation of this spacetime, and it has unacceptable physical properties, for the reasons described in your link.

This is refered to as a ``mirror wormhole''

He's using this term only to describe the nonstandard attempted interpretation.

Does this means that the future singularity of the photon emitted in I is the past singularity of the photon emitted in III?

In this interpretation, the two singularities are considered to be the same, and regions I and III are considered to be the same.

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