These photons, electrons and protons collide with particles in the atmosphere and produce cascades of other particles by converting the kinetic energy of the incoming ray into the products. The cascades happen because the first products are energetic enough to, themselves, cause similar collisions further down the atmosphere. The cascades are seen by flying or land-based detectors.
At the LHC you have two high-energy proton beams colliding head-on. There is a cascade but it is a lot cleaner because there is nothing else for the products to collide with, so they just fly off and decay and the collection of decay products is seen by the detector array.
The LHC is a proton-proton collider. When a cosmic-ray proton hits a hydrogen aton in a water molecule you have essentially the same proton-proton collision. We have met the enemy, and he is us — Pogo
The Supernova accelerated ones are the relevant for the comparison with LHC.
The really high one, e.g. those which are watched at Auger have most likely their origin in active galactic nuclei. Der Amerikaner ist die Orchidee unter den MenschenVolker Pispers
The highest energy cosmic rays should interact with the cosmic microwave background radiation (CMBR) if the energy of the proton is about 6×10^19 eV (in the rest frame of the CMBR) into a Delta+ resonance. This is called GZK cutoff.
The average "lifetime" of a proton for flying without interaction at this energy is about 160 Megalightyears.
A Japanese experiment (AGASA) claimed to have found some of this very high energetic particles, which led to speculations, despite the big systematic uncertainties.
Auger (south, in Argentina) recently reported (maybe preliminary) to have found as well some, but a) there is a reduction in the current on these very high energetic particles b) they were able to track much of the particles origin (at this high energies the intergalactic magnetic fields don't bend the trajectory too much) back to known active galactic nuclei (AGN). AGNs are likely huge black holes (as we have likely a smaller one in our galactic center). On their accretion discs many light year long streams are accelareted to produce high energy particles of all types.
The surprise about their existence though shows only a lack of our understanding of the AGNs, not of the special relativity and not super heavy BigBang relicts, which would have been the most interesting for particle physics. Of course this is not yet completely safe, as Auger is only shortly in operation and it might be that some have different origin, but I wouldn't bet against it. Der Amerikaner ist die Orchidee unter den MenschenVolker Pispers
Which is rather a pity, because GZK violation was one of very few experimental results not explainable by the standard model of particle physics and the standard cosmological model (another one being the rotation curves of galaxies). We have met the enemy, and he is us — Pogo
