First a word about what grain direction is
Paper fibers are made up of long strands of cellulose, physically entangled with and bonded to each other in the papermaking process. Depending on how the paper was made, the fibers may be more or less generally alined in the same direction—what bookbinders refer to as the grain direction. If the fibers are more parallel to the short edge of the rectangular sheet of paper, one says “grain short.” If they are parallel to the long edge, one says “grain long.”
Traditionally, eastern papers are made by dipping the mould repeatedly into a vat, each time bringing up more pulp, and shaking it forwards and backwards but not so much side-to-side. Here’s a nice video that makes it look easy as pie (it’s not). This will make the fibers line up along each other (in the case of the video, grain short). Western papers are made by shaking the mould forwards, backwards, and side-to-side; for that reason it usually doesn’t have much of a grain direction.
On a side side note—one YouTube video leads to another—here is the traditional way to pulp fiber for papermaking in Europe (a waterwheel or wind turbine raises and lets fall these hammers into tubs of fiber & water).
When machines make the paper, the pulp travels along a massive conveyer belt, forming not sheets but a very long strip of paper that ends up on a roll. The grain direction in this paper is very strongly alined with the edge of the strip, parallel to the direction it travels.
There are ways for machines to better approximate handmade paper, but we’ll skip those for now.
How grain direction affects the behavior of the paper
When paper absorbs water, whether from atmospheric humidity, glue, or a flood, the fibers swell and expand. The metaphor is a bit of a stretch but you could consider a piece of paper with strong grain direction sort of like a sushi mat. In the way that the mat easily rolls up in one direction and not in the other, so strongly-grained paper will expand and curl parallel to the grain more than perpendicularly. You can prove it to yourself by taking a sheet of paper, measuring in both directions, getting it completely wet, and measuring again after a few minutes. (And actually, in situations when it matters, like lining boxes or putting down endsheets, you might try gluing out a scrap of the same size of the same paper and measuring how much it stretches so that you know exactly how much to compensate.)
I suppose I should mention at this point that binder’s board (millboard in the UK) is essentially wicked thick machine-made paper and has the same sort of grain direction and same resultant behavior.
Why it matters in your book
(1) Paper bends more easily with the grain direction. If the paper is folded and sewn with the grain direction parallel to the spine, it will drape much more easily when the book is open. Why does that matter? Well. It’s easier to read or write in. But when the opening is easy, there will be less stress on the spine, and the book will last longer. Your reader will also be less likely to force the book open farther than it can. Conveniently I only have a photo of a book I’ve made which opens nicely (funny how you forget to document the other ones), but I’ll dig up something to photograph which doesn’t work as well.
N.B. This one does have correct grain direction; the paper is just too stiff to drape at this size. (It’s actually the same paper as in the image above, but the book is a quarter the size.) BUT—something like this may also happen were the grain direction perpendicular to the spine:
(2) If the boards warp, which they might do for a million different reasons we’ll just not touch for now, the book will be much happier if they warp parallel to the spine. Think about a drastic scenario when the boards have curled away from the textblock a great deal: if they curl parallel to the spine, they will probably stay attached to the spine and textblock, at least more so than if they curl perpendicular to the spine. Perpendicular warping would surely cause them to rip away from the spine fabric or leather. Boards protect the book, so once they start becoming detached, not only is it aesthetically a problem but your book is now exposed all sorts of threats.
I’ll leave you with that for now, and add more later about how to determine grain direction in paper and board.