You have stiffened the paper by greatly increasing its bending moment of intertia. This can occur in at least two ways:
1.) When you unfolded the paper, it still had some residual bend which made the paper form a very shallow 'V'. Even though shallow. it is much deeper than the thickness of the original paper.
2.) When you unfolded the paper, it still had some residual bend which made the paper form a very shallow 'V' which was very localized to the area next to the bend. Even if you refold the paper in the opposite way, to take out or minimize the original fold, there is still a shallow 'V'. Even though it might be much shallower than in case 1, it is much deeper than the thickness of the original paper.
In both cases, the increased resistance to bending comes from the new geometry of the paper, more specifially, the geometry of a cross-section of the paper which goes through the bend.
Think of the unbent paper as a beam. Its resistance to bending is proportional to b(d^3), where 'b' is the width of the beam and 'd' is the depth. If you take a piece of 8.5" x 11" piece of paper and lay it flat over a pencil on the table, the paper will flop so that both ends touch the table. The paper forms a beam: 'b' is 8.5" or 11" (depending on how you laid the paper) and 'd' is the thickness of the sheet (say, about one one-hundreth of an inch).
How to improve the stiffness and strength of this beam? Fold the paper, accordion-style, with sharp 1/2" folds. Then lay it across the pencil, so that the folds are perdendicular to the pencil.
Assume that you folded the paper into an accordion that was 1.1" wide (one tenth of the original unfolded 11"), had about 22 folds and was 8.5" long. The 'b' for this new beam is 0.5", which is 50 times more than the (1/100) of an inch thickness of the paper. The new beam is (50)(50)(50)/10 = 12500 times stiffer than the unfolded sheet.
Now, let's go back to your sheet with the one bend. When you folded that sheet, you increased 'd'. The amount is hard to quantify, but I would ballpark it at a factor of 5, even if you tried to straighten out the bend by refolding it the opposite way. Again, guesstimate that 'b' of the localized area affected by the bend is 1/10 of the original sheet width. So the new resistance to bending increased by a factor of (5)(5)(5)/10 = 12.5 times the unfolded sheet.
Forever_a_Newcomer is on the right lines, but it's not like water dissolving salt.
Paper is mostly made from cellulose fibres (depending on the type there may also be filers and glazes like clay). Cellulose molecules bristle with hydroxyl (OH) groups, and these form hydrogen bonds with each other. It's these hydrogen bonds that make the individual fibres stiff, and also hold the fibres together.
Water is also full of OH bonds, obviously since it's H$_2$O, and the water molecules form hydrogen bonds with the hydroxyl groups on the cellulose, which breaks the hydrogen bonds that cellulose molecules form with each other. There are two results from this: firstly the cellulose fibes in the paper become floppy, because their internal hydrogen bonds are broken, and secondly the fibres separate from each other more easily. The combination of these two effects makes paper easier to tear apart when wet.
Most organic materials show similar behaviour. For example cotton is also easier to tear when wet (cotton is also made mostly from cellulose). Also hair becomes floppier and more easily damaged when wet, though the effect is less pronounced because hair contains fewer hydrogen bonds than cellulose fibres.
Best Answer
Paper is a mesh of fibres usually mixed with a binder and some clay. The fibres will in turn have some microstructure depending on their origin (cloth, wood, etc). Ultimately the paper is composed mostly of cellulose molecules.
When you tear paper you are mostly pulling the mesh of fibres apart. If you look at the torn edge closely you'll be able to see the fibres sticking out of it. You will probably also fracture some fibres, and that fracture process will depend on the microstructure within the fibres. The clay particles are small and rigid, and they are likely to just separate from each other rather than fracturing. Some of them will spring off the surface as a fine dust.
Given all this complexity, the chances of you reassembling the torn pieces into anything approaching their original configuration are essentially zero.