The Italian Sparrow shows that hybrid genomes are variable, but with some important constraints.
Hybridization can create new species. Although this process is rare, there have been several well documented cases in birds (see a recent post). One of the best studied examples is the Italian Sparrow (Passer italiae), a cross between House Sparrow (P. domesticus) and Spanish Sparrow (P. hispaniolensis). You can read more about this hybrid species here and here.

An Italian Sparrow (from http://www.gobirding.eu)
Genomic Mixture
A closer look at the genome of the Italian Sparrow showed that it is indeed a mixture of both parental species. But how much is inherited from each parent? And how many hybrid combinations are possible? Do some genes come from only one parent or are the possibilities endless? These are exactly the questions that Anna Runemark and her colleagues tried to tackle in a recent Nature Ecology & Evolution paper.
Compare it with dealing a deck of cards. Each card is a specific genomic region. Black cards represent the House Sparrow and red cards the Spanish Sparrow. The players receive cards from both colors. Some players end up with a winning hand, while some combinations of cards ensure a painful defeat.
Islands Populations
To compare such differently dealt hands, the researchers needed independent populations of Italian Sparrows. They also needed to be sure that there has been no sneaky exchange of cards. Or to put it in genetic terms: no gene flow. Luckily there are four such populations on the islands of Crete, Corsica, Malta and Sicily. The researchers sequenced birds from these islands and compared the genomes with their parental species.
The results showed that Italian Sparrows from Crete and Corsica received most of their DNA from the House Sparrow, while birds from Malta and Sicily resembled the Spanish Sparrow more. There is plenty of variation in the genomes of these hybrids, allowing them to adapt quickly to new environments. The authors state that “these results illustrate how selection in concert with the parental mosaic is able to form unique features in the genomes of hybrid populations.”
Constraints
Some genes, however, came exclusively from one parent. In my card analogy, this would correspond to a situation where you always receive a black queen, never a red one. There are thus some constraints on the formation of hybrid genomes: not all combinations are possible.
The genes that came from Spanish Sparrow affect external phenotypes, whereas the House Sparrow genes are involved in DNA-repair and functions related to the mitochondria (so-called mitonuclear genes). These genes might contribute to reproductive isolation between the Italian Sparrow and its parents.

A Spanish Sparrow (from http://www.hbw.com)
Z-chromosome
Interestingly, the genes that came exclusively from House Sparrow were mostly situated on the Z-chromosome (see here for more information on ZW sex determination in birds). This chromosome might be an important driver in the speciation process of the Italian Sparrow. However, the Z-chromosome is quite peculiar and we should be careful with drawing conclusions (see this excellent review by Darren Irwin).
As I have written before: the Italian Sparrow is a goldmine for geneticists!
References
Runemark, A., Trier, C.N., Eroukhmanoff, F., Hermansen, J.S., Matschiner, M., Ravinet, M., Elgvin, T.O. & Saetre, G.-P. (2018) Variation and constraints in hybrid genome formation. Nature Ecology & Evolution, 2:549-556.
The paper has been added to the Passeridae page.