I was having a discussion with a friend tonight. He was making an analogy that involved all mules being sterile. I mentioned that I heard here of fertile mules. He said that he’d be very interested to hear more. So, I ask for your help and information.
With DNA testing being increasingly easy to do, cases of mules producing offspring are becoming easier to prove (or disprove!). Hereis an abstract from 1985 in which a female mule (cytogenetically confirmed) gave birth to a male foal sired by a donkey. Hereis another excerpt from 1988 Journal of the Royal Society of Medicine. Googling “mule fertility” pulls up quite a few journal abstracts as well as news reports.
From what I’ve read and heard (which isn’t much…my last genetics class goes back nearly a decade ago), only female mules have been confirmed as fertile; it appears the males produce underdeveloped sperm, though there might be cases I simply haven’t heard of.
Generally speaking, hybrid infertility is more severe in the heterogametic sex - XY instead of XX (Haldane’s rule). In mammals, that means that male hybrids are usually less fertile and/or viable than females. Indeed, there have been no reports of fertile male mules, but a few fertile female mules.
I don’t know if much is known about the genetic basis for the horse-donkey speciation. They do have different numbers of chromosomes - 64 for the horse and 62 for the donkey - which is going to cause problems. Mules have 63 chromosomes, which, being an odd number, means they’re not going to pair correctly. In the simplest case, fertile females may simply, by pure chance, happen to get the right assortment of chromosomes into their ovum without the help of homologue pairing in meiosis I. These rare eggs could then be capable of being fertilized and growing, barring any other prohibitive interspecific genetic interaction. I just don’t know enough about the details of this particular situation to say for sure what’s going on.
When we bought a mule last summer, she was part of small herd of mules, living with a male donkey.
One of the mule had given birth to a young (a male, which looked like a donkey, and was already sold), and since the guy who was selling the mules was totally clueless about them, he had no idea it was very unusual. This mule was pregnant again at the time we visited them.
We almost bought her, for the rare factor, but decided against it, because I had searched the internet a bit and read that since she already had a foal, for the second one unless you were with her for the birth to prevent the foal from drinking the first milk or something, there was risk of death for the young.(plus I didn’t want to have to deal with a barely tame mule and a young one)
I wasn’t able to google it again, so if someone has more info about it…?
For the purposes of the OP’s friend’s analogy, I will submit my own:
A fertile mule is like a four-leaf clover or like a male calico tomcat … rare, but not completely unknown. It does happen, very occasionally.
Whoa, back up here a second! If it is correct that all mules are sterile, then where do baby mules come from? Did they change the rules of Biology since I’ve been in school??? 
Mules are crossbreeds from a male donkey and female horse.
Whereas the rarer hinny is the reverse.