Learning a piece: hands together or separated?

I remember reading about an interesting technique for learning Bach...learn from the beginning HS, then when you have both hands solid, learn it from the end HT, meaning practice m32 HT, then 31-32, then 30-32 etc... it helps avoid the whole 'the start sounds better than the end' thing...I've tried it a couple of times, certainly seems to work!

I generally start learning a piece hands together, but I always break it up later in the learning process, section by section, hands separately. I find it really helps to consolidate the learning and make it more secure. 

I know this might be a bit of a controversial topic, with lots of different opinions, but I generally learn hands together first and break it into hands separate if there's a particularly difficult spot. I generally try to get hands together as soon as possible, though. I don't have any studies to back it up, but I've noticed that my piano students (mostly late beginners) struggle a lot more when I've had them learn hands separate first. 

My current "method to the madness" is pretty in line with what others have written so far, and really does depend on the piece.

I gravitate toward learning hands-together when 1) the harmonic direction is particularly important, 2) if there's a lot of hand-crossing or note-splitting, and 3) generally if the piece is not nearing the limits of my current ability. (e.g., Liebestraum 3, Un sospiro, Ravel Sonatine, Berg sonata).

I find hands-separate then hands-together to be more helpful for contrapuntal/polyphonic works like Bach (which I admittedly don't play that often because I really struggle with it). Also for etudes and other passages where one hand is clearly more challenging to learn (e.g., Winter Wind, Double-thirds, Revolutionary), I definitely spend more time working hands-separately.

If I'm going to be performing from memory, I will feel more confident if I'm able to play through the entire piece hands separate from memory. I've had enough traumatic memory slips that it's worth it to me to put in that extra time/work whenever possible. Otherwise I would perform with the score.

Great question, and very interested to read everyone's experiences!

For me it depends on the difficulty of the piece.

If it is something easier I play both hands to make sure I practice the feeling of the piece.

More difficult pieces I do seperate and recombine, learning couple bars/lines each time.

Fugues and other polyphonic pieces I learn one voice at a time.

I wish I would do more hand separate exercises. I often lack the patience to do it in the beginning. I agree that polyphonic work is super helpful. But many times I found my playing sloppy when I don't do it and have to go back and work hand separate to fix the problem. I wish I have the discipline to practice it that way from the beginning. 

As others has already said, it depends on the piece and passage. Ultimately you'll end up doing both things.
One thing to keep in mind is that you don't have to learn a piece in a linear or chronological way. So you have to adapt and make decissions according to whatever the score is asking.
However, the best thing you can do is always knowing exactly what any hand is doing independently so you can focus on whatever issues may show. But that doesn't mean that you have to always learn them separately.

It’s difficult to give a general answer. I find it important for good fingering to feel how both hands will work together but separate practise is of course often necessary. However, now, the more I can hear in my head, the easier it is to practise with both hands. The less I can hear passages in my head the more analysis and detailed work I find myself doing. 

I know HS is best but I find LH only can be boring to practice so I record the RH and play the LH along with the recording.  Creating a good RH only recording requires practice too so it's a bonus because I have a motive (and a motif :-)