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Let me quote Wikipedia.org:
"The two strands of DNA run in opposite directions to each other and are thus antiparallel."
There is a practice or method used for error correction with regard to parking tickets: First the number plate is written down from left to right, then again but in reverse (right to left) order.
So my question is: to what extent does the antiparallelity of both DNA stands constitute an error correcting code?
Double-stranded DNA does have redundancy giving a basis for error-correction, but it has nothing to do with the antiparallel nature of the DNA.
Each strand has an orientation, that is the pieces of the backbone are not symmetric and so chemical reactions have to happen in one of the two possible orientations along the strand. One end of a strand is called the 5' end and the opposite end of the strand is the 3' end. The two strands of a DNA molecule run in opposite directions (antiparallel), with the 5' end of one strand being adjacent to the 3' end of its partner strand. The two strands do not contain the same codes in opposite order like your license-plate coding example.
The redundancy is only pair-by-pair. A nucleotide in one strand is paired with a unique complement nucleotide on the other strand. It is thus redundant like bit-for-bit odd parity is redundant. This redundancy allows errors to be detected from a mismatch in what should be a pair, and a damaged nucleotide can be replaced with the correct nucleotide by using the complement of the nucleotide on the other strand.
As you have intuited, complimentary base pairing gives DNA molecules 2-fold redundancy. That is, given the base present on one strand, we can always deduce the base that should be at the corresponding position on the other strand.
Cells take advantage of this redundancy to perform error correction through DNA repair processes. Here is a textbook section that introduces how DNA damage is sensed and how cells can use the complementary base on the other strand to guide repair. It also details how cells can determine which base is "correct" when a mismatched base pair arises during replication.
