Am wondering what I should do with our palm that has been damaged from heavy snow? Do we cut off all the droopy leaves and leave the upright? Or cut the whole thing down to the stem?
It looks like a Cordyline to me, and you could hang on a bit and see if the hanging leaves are actually dead or not. If they are, I'd cut back just the dead leaves. Cordyline over time naturally develop a long bare trunk with a crown of leaves. I'd also recommend what you've probably thought of already--when snow is predicted, cover your plant with a bit of fleece.
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Here's a helpful video:
https://www.gardenersworld.com/how-to/grow-plants/how-to-grow-cordyline/