Easiest way is to put the cuttings in a moistened plastic bag and keep in a warm place with good light (but never direct sun). Let the plant sort itself out (it made the mess afterall, LOL).

Yeah the root end should be down.

Tropic's idea is a good one, another would be to just float to pieces of stem in a bowl or tub of water and as new growth emerges pull the sections out and reset them with new new stuff pointing up.

Orientation doesn't really matter for this particular vine, and many others. It sounds like you may want to put these horizontally, when they're ready for a pot of "soil" to get the highest number of growing tips. Each node that takes root is also likely to make a new growth tip, unless it is buried.

Similarly, when the tip of a vine is removed, it has no choice but to make new tips. The pic below is Epipremnum (Pothos) but the principle is exactly the same. The blue circle is where the tip was cut. The red circles are around the resulting new tips.

I have grown them upside down before. They will eventually right themselves with time. The trick is to get one root node

Dellis, thanks for clarifying.

"... lay the section of vine horizontally on soil or float it in water you may get either or both new growth and roots from each node." Well said.

That's what I was getting at. This should happen so fast that direction is not a factor. There's pics of some Philo props here.

It's not been said specifically here that cuttings should have at least 1 node submerged, regardless of where a cut was made. New roots can only grow from a node, not the cut end. If that basic assumption is not known, this could be a very confusing discussion. Whatever size the cuttings are, I would devote any nodes without leaves to making roots, then decide what to keep when potting, if starting in water, see what each one does.

I knew the node rule. I've rooted one or two of these guys before, just not so many from such a messy plant (every time the vines kinked it looked like the leaves changed growth direction, but that's impossible-right?). I think they're going to be okay since none of them withered over night and they're actually starting to look a little happier than they were yesterday morning when I picked the plant up. I'll put up a picture when I have something to show you besides a big jar of leaf spaghetti (like, I don't know, a jar of leaf spaghetti with roots and maybe some token new growing shoots). I hope these guys make it-I have a little heart leaf philodendron I rooted a little less than a year ago that's maybe a foot tall with pretty big leaves and lots of fast growth that I'm growing up a curtain string. That level of health is what I'm going for. Right now...well, these cuttings have got a long way to go before they compare, and as long as those leaves on the cuttings are staying green and turgid for now I'm happy. These little guys seem amazingly tough (though they don't seem to like cold homes, oddly enough).