Recombination on the Y is generally limited to the very tips of the chromosome, which do exchange material with the X chromosome (not mtDNA). The bulk of the Y chromosome (and the place where the genealogical Y-STRs reside) is called non-recombining Y (NRY), and this is passed from father to son every generation.
Women can pass on the exact same X chromosome they inherited from their mother OR their father, but more often it is a combination of the two. Thus in most cases, each child will inherit a somewhat different X chromosome from their mother, a mixture of the X's from their maternal grandparents..
Fathers have only one X chromosome to pass along, so sisters will inherit identical X's from him.
This is easier to see with the animations at SMGF:
You sound very knowledgable, & I appreciate the feedback - but, quoting from the website you referenced:
"Mitochondrial DNA (mtDNA) is a type of DNA that is carried by both men and women but is only inherited from their mother. **Mothers, in turn, inherit their DNA from their ~mothers~** ... and so on back in time along one's maternal line.
"Note above that there is **no contribution from your paternal line, or from any other female ancestor other than ones in your ~direct maternal line~** (as shown below)."
And it has a nifty chart proving what I said in my op. I realize women inherit an X chromosome from each parent (which is why girls may have physical characteristics from both sides of their families), but I was talking about mitochondrial DNA which is passed on to a woman's own children through procreation (not the X chromosome donated from the father to form a female child) & this corroborates what I said: that women normally only ~transmit~ (through procreation) their OWN mother's DNA to their children.
That is the main reason why mtDNA is so useful in population & migration studies. It is also the reason why, in hybrid mammals the mtDNA of the superior species is always transmitted to the (fertile) offspring. Only the female of a superior species may produce ~fertile~ offspring. For example, mules always have the mtDNA of mares (horses), not of their sires (donkeys).
Also, ftDNA states that, "2. The **mtDNA** test on the other hand is tracking DNA that is passed only through the Mother's line and **cannot be passed on through the male line**."
So again, while females carry an X chromosome from each parent, giving them combinations of physical characteristics from both parents, they only transmit their ~own~ mothers' mtDNA (iow, the X chromosome they inherited from their ~direct~ maternal lineage), when procreating.
And it doesn't normally recombine; I've been reading up on that very subject earlier. It seems that only under some very rare, anomolous circumstances (in very, very rare & possibly sterile individuals) - it might recombine a little bit, but not in a way that may be passed on to future generations. But good luck finding such cases, there aren't very many of them. Here's a case where a man was found to have combined mtDNA in his muscle tissue cells only (nowhere else in his body):
And while males typically always carry both their mother's mtDNA (the X chromosome inherited from his mother's ~own~ mother) & their father's yDNA (from his father, but recombined slightly somehow, as you pointed out), giving them some combination of physical characteristics from both parents, they naturally only transmit their yDNA to their sons.
That explains how males may transmit family traits from females of their lineage, to their sons: through recombination. Whereas, although women may express some of the physical characteristics of some of their father's female ancestors - they may only donate the X chromosomes from their *direct maternal lineages* to their own offspring.
In my op I stated that mtDNA doesn't recombine, whereas yDNA does recombine, & I stand by that claim. I was referring to the type of natural Human procreation that generally produces fertile offspring.
I believe your statement, that women may pass on either X chromosome to their children (her mother's "OR" her father's) is generally erroneous. Mothers don't normally combine the two X's either, before transmitting their mtDNA to their offspring.
It seems biologically fair, since the chromosome(s) donated by fathers depends upon the genders of their child(ren). Fathers normally don't combine their two sex chromosomes (X & Y) before passing them on to future generations, so why would women?
Additionally, men may potentially sire hundreds or thousands of offspring; but women are biologically fit only to have a few babies in a lifetime (unless you're octo-mom, lol). Even 25 kids (like the amazing Arkansas woman) is nothing, when compared to the number of children a man might produce.
(However, imo the number of children produced isn't as important as the number which survive & thrive.)
BTW, I watched the short but very impressive animation on Sorenson's Molecular Genealogy Foundation which you referenced; and it unfortunately does add to the confusion, by contradicting themselves in their own statement which I quoted for you above (that I got from the text section of the same website, under the title, mitochondrial DNA). I believe it's the animation which is flawed, not the text (based on all of my other research).
The animation cites no scientific literature to back its claim.
UPDATE: Here, I was trying to hash out the finer details of inheritance with regards to mitochondrial DNA and the sex-linked genes (Y and X chromosomes). The participants on the DNA Forums who were self-proclaimed geneticists were not being very kind or patient with me. They didn't do much to shed light on the subject either, which meant that I had to figure most of it out all by myself, with my own research.