Is photon emission the time-reversed process of photon absorption? Say we have an atom on which we shoot a photon. Is the process of absorption the time reversed process of emission? I can't imagine the two processes being the same, although in both cases the photon has the same mean  energy. Can we tell, if we "see" a photon coming out of an atom, if time is going forwards, or if it's the time-reversed image of absorption?
In Compton scattering, the two real photons, before and after scattering, have different energies, but still, the process could go as well forward as backwards in time.
On the other hand (say in a hydrogen atom), if we consider the absorbed and emitted photon, they have the same energy (momentum) but I don't think both time directions are the same, like in Compton scattering. So, contrary to Compton scattering, we should be able to see the difference between:
-a photon being absorbed and subsequently emmited spontaneously
and the time-inversed proces:
-the emitted photon becoming the one absorbed, and the absorbed photon the emitted.
Of course, absorption is a reversible process, like opening or shutting a door (you can see a difference though if we "play the film in reverse", so you can tell if time goes forward or backwards. Can we say in which direction the clock ticks if we "look" at an absorption-emission process (unlike the scattering of a photon with an electron)?
 A: Absorption is a word used somewhat ambiguously. In the case of Compton scattering we are talking about an elementary process, governed by a certain Hamiltonian and reversible equations of motion. So this process is reversible. On the other hand, in case of an emission/absorption by an atom we usually mean it changing irreversibly from one state to another. If we were to treat emission by an atom by exactly solving the Schrödinger equation for an atom coupled to electromagnetic field, the atom would continuously reabsorb and reemit the photon into different modes of the field. In practice this does not happen for the reasons that are usually not discussed in simple treatments of emission and absorption, but do introduce irreversibility: infinite number of the radiation modes, coupling to the environment, finite experiment duration, etc. See, e.g., my posts about what is hidden in Fermi Golden rule (this answer and the answers linked in it).
Remark: Janes-Cummings model is an extreme example where the atom is coupled to only one light mode, and the reversibility (in the form of Rabi oscillations) is manifest.
Book recommendations:

*

*Rodney Loudon, The quantum theory of light.

*Claude Cohen-Tannouji et al., Atom-Photon Interactions: Basic Process and Applications.
