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I'm talking about various forms of infinitesimals, yes. In a sense they're zero, but in another sense they're not.

For the reasons you mention, these models have 0 != -0. :-)

But don't these models all have a true zero, distinct from the infinitesimals, which is its own negative?

Usually, yes. But sometimes it's written as 0^\infty.

"Usually" So there is an instance where this isn't the case?

Well... notation. People come up with all sorts of ways of writing things and all sorts of weird models.