Are there any known real-world instantiations of additive groups/rings in which addition is rightward elongation of a left-to-right ordered string and subtraction is leftward elongation of a left-to-right ordered string ?
In this regard, note that:
1) duplex DNA, i.e. the classical "double-helix" type of DNA, has two ANTIparallel strands, meaning that:
a. each of the two strands has a 5'-3' direction and a 3'-5' direction
b. the 5'-3' direction on one strand is the 3'-5' direction on the other, and vice-versa, of course.
2) any segment of classical duplex "double-helix" DNA can be elongated IN EITHER direction, i.e. if we pick a direction on one strand, say the 5'-3' direction, then it is true that the duplex itself can elongate in this direction, or in the opposite 5'-3' direction on the other strand
3) therefore, if we consider a duplex segment as a WHOLE (not one or the other of its strands), then the segment can elongate in either one of two directions.
Also, it is probably important here to note that there are species which exhibit closed "circular DNA duplexes", analagous to projective lines vs affine lines.