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Tyrrell_McAllister comments on Why is the A-Theory of Time Attractive? - Less Wrong Discussion
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This is what I was referring to. The axioms of ordered geometry, especially Dedekind's axiom, give you the topology of the affine line without a distinguished 0, without distinguishing a direction as "positive", and without the additive structure.
However, in all the ways I know of to construct a structure satisfying these axioms, you first have to construct the rationals as an ordered field, and the result of course is just the reals, so I don't know of a constructive way to get at the affine line without constructing the reals with all of their additional field structure.
You might be able to do it with some abstract nonsense. I think general machinery will prove that in categories such as that defined in the top answer of
http://mathoverflow.net/questions/92206/what-properties-make-0-1-a-good-candidate-for-defining-fundamental-groups
there are terminal objects. I don't have time to really think it through though.