This is not limited to photons, although the first part, "no mass" is due to the rest mass of photons being zero.
Two objects of any kind whatsoever have more mass going in different directions than going in the same direction.
You are right, I was sloppy with language, and accept my downvote honourably :-)
I was talking about what is sometimes called relativistic mass or effective mass (which are considered obsolete terminology), in order to address the GP (brabel)'s belief that photons having zero mass will have zero gravity.
> since the same sort of sources that teach you about these concepts are very emphatic about photons being massless
It depends how old you are; I learned these concepts in the 80s :-)
From https://physics.stackexchange.com/a/133380:
> There's no controversy about whether mass increases or not, there's controversy about what you call mass.
Something that may help readers:
From https://www.desy.de/user/projects/Physics/Relativity/SR/ligh...:
> Does light have mass? The short answer is "no", but it is a qualified "no"
> Part of this discussion is only concerned with semantics. It might be thought that it would be better to regard the mass of the photons to be their (nonzero) relativistic mass, as opposed to their (zero) invariant mass. We could then consistently talk about the light having mass independently of whether or not it is contained. If relativistic mass is used for all objects, then mass is conserved and the mass of an object is the sum of the masses of its parts. However, modern usage defines mass as the invariant mass of an object mainly because the invariant mass is more useful when doing any kind of calculation. In this case mass is not conserved and the mass of an object is not the sum of the masses of its parts. Thus, the mass of a box of light is more than the mass of the box and the sum of the masses of the photons (the latter being zero). Relativistic mass is equivalent to energy, which is why relativistic mass is not a commonly used term nowadays.
Here is one of the 3-5 interesting facts I remember from my modern physics class: Two photons going the same direction have no mass, but two photons in different directions do have mass. Unfortunately I don't remember the math well enough to explain it. Maybe this means your right about being able to put photons in orbit around each other.