The presumption that humans and Neanderthals interbred is gaining ground among scientists, despite the two lines having a different number of chromosomes (this is not yet absolutely proven to be scientific fact). Horses and donkeys interbreed to produce mules. Occasionally a few of these mules are fertile and can have offspring, so a different number of chromosomes is not an absolute barrier.
Neanderthal DNA is present in humans. However, there is no mitochondrial DNA: this is passed along the line of the mother. Visual evidence of interbreeding can be observed. Tatjana Schmidt-Derstroff noted decades ago that there are two types of Australian Aboriginals, desert living long-legged gracillis and coastal living muscular robustus. Both are dark skinned. Robustus carry a red/blond haired gene (not passed on by shipwrecked European sailors). The more muscular type is roughly similar in features to Neanderthals. For the most part these two different kinds of Aboriginal remained separate, though some interbreeding did occur. Today, most Aboriginals have part European ancestry.
Tatjana Schmidt-Derstroff is happy with the hypothesis that humans and Neanderthals had fertile offspring, along the male line of course. This would engender a male Neanderthal mating with a female human. With male offspring passing on respective genes. Ms Schmidt-Derstroff also holds that the major leap from Neanderthal to human is about to take place again to modern space-travelling humans. Well, this change is already taking place as people move to new countries and different "races" interbreed to create a new tan colored human.
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Anthropology