Four years ago, I got this house plant, and another identical one. the leaves are about 6" wide by 10" long.
They have grown up from a reasonable height to taller than me now, and they are tipping over. As you can see, they’ve grow straight and very top heavy.
I’ve tried to look on this online, but identifying plants isn’t as straightforward as it would seem.
Part two. Can this plant be cut and replanted shorter?
Thanks!
TP
From your photo TPlayer, I can’t give you an ID, but the general rule for plants like this is that they do very well to be knocked back via pruning, AND that the cuttings you get from the prunings will probably be super-simple to strike as well…giving you even more of the monsters to deal with in the future.
Good luck 
There are several interactive plant identification guides online. You need to describe the stem, the leaves, the fruit, type of root, etc.
I’ve used several to identify trees, anyway.
Have you links for any of them?
Thanks!
I’d imagine that doing an air-layer type thing would have a good chance of success to make more of that plant (if you are committed to the process). Here is a quick explanation with a not-so-good graphic linked in it, too. Basically, you cut/scrape into the side of the trunk (to growing-layer/xylem only! just below surface tissue usually) at level where you want new roots to grow, or at height of where you want to it to be when done. Wrap sphagnum moss or other moist ‘media’ around trunk and then hold in place with plastic wrap (saran wrap is perfect, ime). Keep the moss barely moist (to keep rot from happening) and after a few weeks, more or less, roots should start forming/growing into the moss. When roots are almost filling the wrap, cut off below the roots, and plant in a new appropriate-sized container. Don’t overpot (use a planter that is over-sized for root-ball size) to help keep the roots from rotting, and repot as needed 
Enjoy the new plant, and the old trunk will likely bud new growth, possibly even before you get the air-layer portion cut off. I wish I could help ya on what species that is, but not familiar with it. Also, google ‘air-layer techniques’ (or ‘air layer-plant cuttings’) for graphical instructions, etc - its pretty common, and really easy, thing to do with plants, fwiw.
It looks kind of like an Abutilon, but I have no idea which species.
Seconded. And great user name for participating in this thread.
Thank you. Googling, it says that this technique is used if the plants don’t root easily. Do you think this is the case for my mysterious houseplant?
This is a start. Thanks.
Air-layering being a method that takes a bit of attention and labor, its usually ‘reserved’ for those species that don’t root from the easier methods. I mentioned it as an option as I do not know if other methods would be worthy of attempt. I wish I could help you more, but my ignorance of plant species (or type overall) keeps from giving more focused advice. I have rooted LOTS of various species, from near-impossible cultivar-type red maples to whatever else I had around me. Unless you know the species, knowing exactly what method works best (or even with high-% success) is hard to say. I be surprised if there was no regrowth if you just chopped the trunk way down, but not the best approach not knowing its tendencies, of course.
Air-layering is good when loss of leaves and lack of vigor for new growing ones is an issue as the roots are already in existence when the cutting is done. Much less ‘stressful’ upon the already-existing leaves and allows more vigor overall.
Also, a plant that is spindly of trunk (not able to support itself unless its a vine) but large of leaf is usually one that is wanting more sunlight; no idea of plant’s location, though 
Only kinda.
A lot of these house plants are rainforest understorey species. They live quite happily in that suppressed form, but if they get a break in the canopy they will grow into a tree very rapidly.
So while they may technically want more light, if you give it to them your houseplant will try to grow into 10-15 metre tall plants. This plant looks like it may well be one of those. It certainly doesn’t do them any harm to live in their supressed understorey form with the large leaves.
I totally agree with everything about the understory plants and all - but those typically do not grow tall then start falling over (as OP is stating). I was just saying it could well be lack of light that is causing it to be spindly (or have inability to maintain its vertical nature) 
Adding some light could give a ‘better-looking’ plant, but might have to be pruned more often, too, LOL.
These are getting plenty of sunshine, although not direct light. See here.
They just grow up, get new leaves and lose the ones below. The trunk(?) remains very thin.
Diffenbachia act exactly as you state - almost like a vine with limited side ‘branches’. A diff does not really branch much at all (in a home) for the most part. It excels at extending its length by adding new leaves as it drops the lower ones. They are VERY easy to cut shorter and replant, ime, and I have had a few that exceeded approx 12-14’ of length with trunk about size of average male wrist or so (smaller than you’d think probably) in just a few years. They don’t require direct sun, but will happily tolerate direct sun when acclimated properly.
Overall, a really easy species that handles ‘abuse’ such as bad watering habits or bunched roots somewhat well. I’m presuming it is a diff and that you looked at some other images? Especially the trunk and how it usually has a bit of old dried leaf base hanging at various places (at the ‘joints’ of the trunk, so to speak)… IIRC, it has kind of a white sticky sap (??) that is best not touched (keep outta the mouth!)