Here some not too well structured thoughts ...
As you can see in the following link:
http://hcc.web.cern.ch/hcc/beam/by_beam.php
the "crossover" points are located at the experiment points (Atlas, Cms, Alice, LHCb).
They are not actually showing as an "X" layout you can see crossover at one point, but in the experiments (imagine the detector size, so we are talking about meter scale here), the beams run through one pipe in opposite directions. Hey, they are running with nearly the speed of light in practically perfect vacuum, so it's not too difficult for each bunch of protons to do this.
Remember the LHC's predecessor, the LEP (Large Electron Positron collider) that was housed in the same tunnel, was running the two beams in the same (oval shaped) pipe. They could do this because the beams of electrons and positrons have different electric charges, so a magnetic field will force the particles in opposite directions. The size of the particle bunch is order of magnitude mm, the beam pipe diameter cm. Now the magnitude order for the area is squared ! So there is lots of room for two beams in one pipe without "hitting" each other (or the wall of the tube, as the beams tend to "wobble" around there trajectory and aside the dipole accelerator magnets, there are also quadrupole and sextupole magnets to keep the beams focussed and on track. Just search the net for quadrupole + lhc and you'll find some interesting). Getting off-topic here ... back to the LHC ...
In order to be able to get to higher energies, the LHC uses protons that are much heavier, hence at the same speed have more energy (E=mc²). But to accelerate these in two opposite directions, they need two opposite oriented magnetic fields, hence the two tubes needed.
PS: it is rather difficult to exactly "align" the proton bunches and let them collide at the right spot, and even then, you have to imagine the proton density as swarms of mosquito's flying in opposite directions through a giant tube: the chance two "protons" actually interact and create some interesting new particle is still "small", if it wasn't, after a few cycles, all protons would have all reacted with each other and the beams would have vanished.
PS2: now try not to flip: the "low density" of protons is still resulting in 40.000.000 collisions per second (just multiply the bunch density with the number of times bunches crossover with speed of light, a 27km tunnel is traveled many times per second !), and then realize every collision generates megabytes of data, so we are talking of petabytes of data each second. of course even with the grid it is impossible to store and process all this data, so the experiment will filter out of the 40.000.000 collisions a second the 100.000 collisions of data that show one of the predefined "interesting" signatures, and this in 25 nanoseconds time; the 100.000Hz can be studied for a full second or so and then only the 100 most interesting of these are kept and put on the grid.
Thats my understanding of what happens ... feel free to correct as I'm not a professional physicist at all (but an electric engineer with a big interest in physics and mathematics)