WinSCP appears to be setting the modified time of the remote file to my local system's timestamp. Is there any way I can get it to not mess with the timestamp at all? (That is, when it writes the file, the last modified time will be whatever the server assigned it?)

WinSCP sets the timestamp of uploaded remote file to match timestamp of local source file (match in terms of UTC time).

To make WinSCP not to update the time, i.e. to keep whatever timestamp the system assigns to the file, uncheck Preserve timestamp transfer setting:
https://winscp.net/eng/docs/ui_transfer_custom#common

Ah, thanks. The name seems kind of counterintuitive to me, but I couldn't for the life of me find the checkbox you were referring to until I finally thought to click 'edit' on a specific entry in the transfer presets. Any chance of a global option that will apply to the whole connection so it doesn't need to be changed in multiple places?

It is a global. Or what multiple places do you mean?
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Martin Prikryl

I mean, you have to set it for all the transfer setting presets and the default, which is 4 dialogs

Ok, I see. If you actually use (do you?) all the presets, than yes.
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Martin Prikryl

I don't use any of the presets, actually, but they're already there. I don't really know what effect deleting them would have, for example on text file translation between linux <-> windows, or other subtle things I may not know about. Actually, I was used to the one-window-per-session layout so I didn't realize that the presets were global until you mentioned it, so thanks