First off, the type of cards you are using makes an enormous difference. Cheap, (typically) Chinese-printed cards most of the time are not physically capable of faroing due to the way they are made. United States Playing Card company's brands are usually good for faros, but lately I have purchased some cards from supermarkets that will not do it at all; namely, Bicycle Plumas and the green and purple peacock-themed Bicycles. I also have purchased Bees recently (again from a large chain store) that will not faro at all.
Another factor that has an influence on your faros is which direction you do the weave. Generally, USPC decks will faro from backs to faces - I find that quality cards will usually weave either way. Another very important tip: always use a deck with very smooth, worn edges to practice with. The shuffle is much easier with worn cards.
What you're talking about with the weave generally has to do with the cut. More on that in a moment, but first, when any deck capable of weave shuffling is used often enough in a variety of ways, it can do so in both directions. It's simply a sign of the cards being well broken-in.
When a deck out of the box weaves from back to front, that's a "modern cut" or "face-up cut" deck. The deck is pushed through the cutting die with cards face up so the cutting edge enters from back to face. If you attempt a table weave shuffle, it will only do so if face-up.
A "face-down cut" or "traditionally-cut" deck pushed cards face down into the cutting die, entering face to back and reversing the bevel of the card edge compared to modern cuts. These can not only be table weave-shuffled face down, but also can be controlled better in a variety of shuffles fresh from the box.
USPC for the most part stopped making traditionally-cut, mass-produced cards sometime in the 1980s - cutting face down requires an extra step in the printing process where the deck sheet has to be flipped over. Makes little sense to me, since you could simply print the backs on the faces side and vice versa, but I'm not a printer so maybe I'm missing something. Any deck designer who asks them for traditionally-cut cards will often be told that it results in a rougher cut - I call bull on that, since that's more likely caused by a dull cutting edge on the die. You do occasionally find a traditionally-cut deck (could be by accident) in a recent mass-market print run, but even if this happens, it won't be in many of the cards, assuming they catch the mistake right away - a roll on the web press only produces about 11,000 decks and they'd know immediately when the cards came out facing the wrong direction. There are two major customer types that insist on getting traditionally cut decks - Richard Turner and nearly every single casino they do business with, if not all of them. Bill Kalush did as well, back when he was still ordering from USPC - all CARC-produced decks are traditionally cut.
I don't know much about caristry, but I do know a fair amount about card manufacture!
A little tip, Ryan - there's two kinds of faro, faro-in and faro-out. A faro-out is when the top and bottom cards remain the same after the shuffle is completed - the top and bottom cards are pushed out in the weave rather than into the deck. In a faro-in, the top and bottom cards change. If you wanted, you could flourish the cards around, performing faro shuffles and some false cuts and leave the deck in a specific state:
26 faro-in shuffles will reverse the order of the cards in the deck from the beginning.
52 faro-in or 8 faro-out shuffles will bring the deck back to the original order you started with.
Many decks are released in what looks like a random order out of the box, but after a specific faro (usually a faro-out) and maybe a specific cut, the deck will end up in a specific memorized stack order, usually the Mnemonica stack.
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