I read that an LED (primarily emits light) can also be used as a photodiode (primarily detects light). Is this true in reverse? I mean, can a photodiode also emit light?
Yes it can, and does. Any recombination of the carriers (electron and holes) which happens all the time will emit light with an energy level corresponding to the bandgap, 1.12 eV (minus just a little bit) ~ 1 um for Silicon. Although, it must be noted that this is a very inefficient process. It is not used directly because of that but is used in some diagnostic tools.
A direct bandgap photodiode can certainly emit light, and the amount of light can be modulated by loading it differently.
Shine light above the bandgap energy on a III-V junction and you will make carriers. If you have left the leads unconnected rather than reverse-biased it, then those carriers may spend enough time near each other so that a significant fraction will recombine with each other, rather than at non-radiative sites such as defects and dopants.
In effect, photodiodes will "fluoresce" at the bandgap energy. Short it or reverse bias it, and you can collect more of those e-h pairs and the glow (during illumination) will be reduced.
It doesn't really matter if you create e-h pairs electrically or photonically, once they are there, a fraction will undergo radiative recombination.
See for example the answer(s) to Do III-V based photovoltaics “glow” (photo-luminesce) when illuminated but not loaded?