(another egg got the pair of 'mama', above).
Actually, Ovogenesis results in a single egg and three much smaller cells called polar bodies. The polar bodies just die off, as all the resources they would need to survive are taken by the egg. There is no egg that gets those specific genes not found in another egg.
Genetic recombination technically does happen with the X and Y chromosomes (Allosomes) too, not just the Autosomes. However, this is limited to the pseudoautosomal regions, which contain barely 5% of the genes on the Y chromosome and much less on the X chromosome.
Genetic recombination can also technically happen even between completely unrelated chromosomes. These Chromosomal Translocations are the cause of many kinds of cancer and several other harmful syndromes. They can also be harmless, in those who still inherited all of the right genes just in unusual places. That does however make it much more likely that the next generation will experience harmful translocations. Translocations that do not effect an individual's health may still cause infertility.
The Y chromosome mutates at a faster rate than other chromosomes, possibly because it is the only chromosome that must always pass outside of the body every generation. The genes in eggs are much better protected from various mutagens, including radiation.
It may also be worth noting that sometimes a genetic mutation can be beneficial to one sex and harmful to the other, and that it is more common for that to mean that it helps women more than men.
The X chromosome contains genes related to color vision. I a man may be red-green color blind or blue-yellow color blind due to a single mutation. A woman who has a normal color vision gene on one chromosome and one of either type of color blindness genes on the other will see color as normal. She has to have
the same type of color blindness mutation on both chromosomes in order to be color blind. A woman who has the gene for red-green colorblindness on one chromosome and blue-yellow colorblindness on the other chromosome is not colorblind either. In fact, her color vision is superior to that of those who do not carry either color blindness gene. She produces an extra type of light sensitive pigment in the photoreceptors of her eyes, and can see 4 primary colors instead of 3.