[whatshisface]

If the emitted photon always had the same momentum as the absorbed photon, the light would go right through the mirror, passing on its way without changing direction.

[samatman]

Only if the momentum wasn't transferred to the mirror.

Which is, in fact, what happens. Solar sails exploit this fact:

https://en.wikipedia.org/wiki/Solar_sail

[whatshisface]

If momentum was transferred to the mirror then the emitted photon would not have the same momentum as the absorbed photon.

[melon_madness]

Conservation of momentum doesn’t mean that the momentum of the emitted photon has to be equal to that of the absorbed photon. It means the sum of the momentum of the emitted photon and the mirror must equal the momentum of the absorbed photon, assuming the mirror had no momentum to begin with.

[samatman]

> Why the photon is then re-emitted in exactly the same direction that it would have if it was reflected rather than a random direction?

That is, the equal and opposite direction, exactly as predicted by conservation of momentum.

[mr_toad]

Momentum is a vector. The sum of the vectors is preserved.