WinSCP can do both:
https://winscp.net/eng/docs/task_keep_up_to_date
https://winscp.net/eng/docs/task_navigate#synchronize_browsing

But actually not at the same time.

Though you can run two copies of WinSCP, one in a background, doing the "keep remote directory up to date" and another one for a real interactive use (browsing, etc).

Is there a way to set it up such that what I do on the local end is mirrored on the remote end?

In other words, if I change directories on the local side, they directory changes to the same one on the remote side. If I delete a file locally, it deletes it remotely.

Thank you for your help, anyone!

-Geoff