You make an incorrect assumption here:
These atoms are obviously split when the paper is torn
When you tear a piece of paper, what is actually breaking are the bonds between different atoms. These bonds are entirely dependent on the configuration of the electron cloud of the atom, and have nothing to do with splitting the atom itself (which, in colloquial language, usually refers to nuclear fission). So, when you tear a piece of paper, the electron clouds of the atoms on either side of the tear rearrange themselves, but by any conventional definition, no atom can be called "split."
Even in cases where an electron transfer occurs across the chain, the resulting process is referred to as ionization rather than atom-splitting. It is quite difficult indeed to be in a situation where there is a significant amount of net ionization on both sides of the tear, so you can safely ignore this for a piece of paper (though it's not insignificant for certain materials: for example, unrolling Scotch tape extremely fast causes massive temporary ionization, and the displaced electrons rocket back to their original atoms, emitting X-rays in the process).
With that in mind, we can answer your question in two different ways. Of course, since there is no atom-splitting going on when you tear a piece of paper, you technically don't need to do anything to "put them back together." But if we assume that what you meant was:
The bonds between the atoms are obviously split when the paper is torn, but is there a way to put them back together?
the answer is yes, because this is precisely why paper recycling works. The incoming used paper is washed, to remove ink and other contaminants, and then left to soak in a particular solution (the composition of which partly determines the color, consistency, strength, etc. of the resulting paper), where it eventually turns into a slurry. Paper is made of long fibers of cellulose arranged essentially randomly; when immersed in water, those fibers spread out throughout the solution. Then the slurry is rolled into sheets and left to dry; as the water leaves, the cellulose fibers end up weakly attracted to each other (the "weak" part is important; it's why you can easily tear a piece of paper in the first place), which leads to a similar kind of random arrangement of weakly-bonded cellulose fibers that we start with.