We have learned that the same information is saved in the DNA in the same place. So when DNA splits and then connects again (during reproduction); how does it maintain the integrity of the information? In other words, do they split and come back like a zipper, or does something else happen?
In the case of replication the parent strands do not pair up back into the original helix. Instead, as your picture and the one below show, the complementary bases are paired up on one of the parent strands, resulting in two copies, each with one parent strand and one new one.
Source: Kathy's webpage on DNA
When I was reading the question, I had a flash of insight into what I presume the question the OP is asking: how do we maintain "integrity" of DNA when we get a set of DNA from mother, and the other set from father (ignoring mitochondrial DNA and other neat things). I would assume this is the question due to the key word reproduction, but anyway,I am here to provide an answer.
As an answer:
I believe the OP has a misconception. DNA of the parents do not pair up in the progeny. Rather, we have 23 pairs of chromosomes, or 46 total chromosomes. What does this mean? The set from the mother and the father do not come together, like a zipper, and have certain bases that just happen to be different (due to single nucleotide polymorphisms and whatnot). Instead, there are two sets of DNA, each with a double helix.
I think this may be what AliceD is also conveying, but I wished to add some more points such as bringing to numbers to help clarify. If I misunderstood, please clarify!