I have 2 pieces of 5-ply birch plywood, and I want to glue them together to create a 10-ply piece.
My question is, does it matter which way I orient the two pieces? Should the grain be perpendicular, parallel, or does it not matter?
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The layers of plywood are already alternated, this is what gives it its strength and warp resistance. Ideally you would place the layers perpendicular, but I don't think you'd see any negative effects if two of the layers happened to be glued together in the same direction.
Plywood is a sheet material manufactured from thin layers or "plies" of wood veneer that are glued together with adjacent layers having their wood grain rotated up to 90 degrees to one another.
If you were to shave off each subsequent layer of a piece of Baltic birch plywood, you would undoubtedly find that the plies are oriented 90 degrees to each other.
My question is, does it matter which way I orient the two pieces?
Absolutely not. Baltic birch plywood is known for its good dimensional stability, which is mostly due to the ply arrangement as discussed. Gluing two separate pieces will not result in any contradiction of grain orientations, thus you will not have any issues with differential shrinkage wanting to pull the piece apart or split it.
No it doesn't make any significant difference which way you orient the two pieces of plywood.
However there is potentially another issue and that is that with some modern plywood the surface veneers are very thin. I see you've specified that you're using birch plywood but that is a very broad category and the quality can vary widely, some types have these extremely thin face veneers. To maximise the bond between the two pieces of plywood you may want to remove that veneer from the face of both pieces prior to glueing them together.
What matters here is what look you need. Like others have said the structural integrity is already in place. With most things you build the grain will look right going a certain way. Sides of a shelving unit the grain should be up and down the shelves left and right and so on. I have made these mistakes when cutting and used the pieces anyway to find out if really makes a difference. So for strength it does not buy for looks I say it does. What are you making?