My understanding is that a photon is pure kinetic energy and has no inertial mass of its own (or probably too low to be significant). But for a box with two photons whose momentums cancel, the photons will affect the inertial mass, because now the energy is stationary when seen from outside.
Except for gluons and photons all elementary particles get their mass from Higgs potential energy.
Is this the only reason why photons don't have inertial mass? That they are purely kinetic and don't interact with any field to gain potential energy?
So the picture I have in my head is that "stationary energy is what has the property of inertial mass and energy that has no form of stationary energy involved has no inertial mass."