Like several on this list, I believe the best way to make relativity perspicuous is to use its absolute-geometric form, rather than its more popular relative-reference-frames formulation. However, it isn't always easy to know how to represent frame-dependent concepts in a coordinate free way, and this is one that I've been struggling with of late.
Just to make sure we are all on the same page, let me declare (quite uncontroversially, I hope) that a massive particle (i.e. one such that m > 0), has a well-defined (time-like) world line in absolute space-time (let's restrict ourselves to Minkowski Space here), and at each event of its world-line we can define (intrinsically - using differential geometry) a unit tangent vector U, from which we can construct the vector P := mU and call it the (absolute) four-momentum of the particle.
But then we come to massless particles - the photon being our prototype, of course. Now without rehashing the details, which have already appeared on this list from time to time and are again bread and butter stuff, the frame-dependent four momentum of such a particle can be defined as (|p|,p), where p is the three-momentum of the particle. But I now want to express this vector as an abstract vector in Minkowski Space, just like I did with mU, and here's where I run into a problem (hopefully, just reflecting my own confusion). In particular, what is the equivalent object here to a 'unit tangent vector on a particle's world-line'? For one thing, the vector has zero-magnitude, which doesn't concern me in and of itself, except I'm not sure then how to write it down abstractly. Also, such an object doesn't seem to be unique. Wouldn't any null-vector along the trajectory work (even if I could specify one, which I'm having trouble doing).
Since none of the standard relativity tomes on my shelf (Rindler, Misner et al, Wald, etc.) seem to describe the idea of an absolute four-momentum vector for massless particles, even though they do discuss photons, I'm wondering if this issue points to something deep. Could the problem be, for example, that there is no such thing as a world-line of a massless particle? That would be interesting, because it would mean relativity is telling us something significant about the structure of the objects inhabiting itself (and this wouldn't be the first instance of this). After all, although we all know that it is impossible to accommodate features like 'interference' of photons in terms of well-defined world lines, how on earth does spacetime know about this??