If you are seeing this effect after the jars have been in storage for a long period, do not eat the contents! This is a sign of botulism due to improper canning; the bacteria often (but not always) produce gas as they grow spores.
If this is happening immediately after the canning process, it is probably because you are not creating a proper vacuum seal. There are three accepted methods for doing this: Thermal exhaust, mechanical sealing (i.e. using a chamber vac) and steam displacement AKA steam injection. See the link for more information; the second two require specialized equipment, so thermal exhaust is what's normally done in a home setting.
In the thermal exhaust method, you get the contents very hot (71-82° C), which causes them to expand and release gases (air and carbon dioxide). After sealing and cooling, the subsequent contraction will create a vacuum seal. This is the most common method of home canning.
If you dump in hot brine and immediately seal the container, you are doing the opposite of this. Since the heat takes a while to distribute, you are causing this initial expansion when the jar is already closed, and this will force more gases up into the headspace and probably pop or warp the lid; if you're unlucky, it might even break the container.
Is this method safe? No. How unsafe it is depends on the acidity. Low acid foods must be processed in a pressure canner. That includes the majority of vegetables and especially garlic, peppers, etc. The main reason is botulism; the anaerobic environment is perfect for the C.botulinum bacteria, but spores cannot grow in high-acid environments with pH < 4.6. If the food is high-acid or has been acidified (i.e. pickled) then you don't need to worry, but with low-acid food you must kill all the bacteria and spores, and botulism spores are extremely heat-resistant.
In fact, they are so heat resistant that you cannot even kill them reliably in boiling water. That is why you need a pressure canner, to get the temperature all the way up to 121° C / 250° F, and you need to hold it there for at least 3 minutes. That is the only way to safely can low-acid food. Simply boiling it isn't good enough, and pouring in some boiling brine definitely isn't enough.
For high-acid foods, such as jams, a hot water bath is OK. This still involves boiling the entire jar after it's been closed, but you don't need a pressure canner. Pouring boiling liquid into a jar of cold or warm food will still not get the food up to a sufficient temperature, and even if it did, you would still need to sterilize the jar itself; that's done by boiling the whole apparatus. In this case you're not worried about botulism (since it can't grow in those conditions), but you still have to kill the other kinds of household bacteria, which don't have heat-resistant spores.
Since canning is intended to preserve the food, i.e. for long-term storage, you have to be a whole lot more careful about bacteria. You can't leave any opportunity for it to grow. Lots of people can the way your wife does and don't get sick, but it is a risky proposition, and I would strongly recommended you use safer methods, especially if guests or children are involved.
I live in Italy - the Quattro Stagioni brand here has been around for more than 30 years, and it is considered the golden standard for in-house canning (other Bormioli glass products are well regarded, too); everyone I know who canned something in his or her life have been using them, and I've never heard of anybody saying anything but great stuff about them. It's rare that I go a whole month without eating something that is coming out of a Quattro Stagioni jar. The one-piece lids are indeed very handy, even if they are a bit pricey (but you probably already noticed that).
All of the Bormioli material (website) and instructions (I'm reading them now from the lid packaging) only mention canning with boiling (pasteurization to be precise), and I've only seen them used that way. This restricts the type of foods you can can (pun not intended) to highly-acidic ones: fruit conserves, tomato sauces, pickled vegetables, etc.
Of course, for maximum hygiene you should follow the instructions to the letter, and use a new cap every time. However, I've seen people successfully re-use caps for canning with less spoil-prone foods (e.g. pickled vegetables), but this is anedoctal and I don't think I can really suggest it.
The lid packaging also report a toll-free number “grandmother Amelia info” (not joking), but unfortunately it's late now as I write; I might call next Monday if I have time.
Happy canning!
Best Answer
I know the inversion method doesn't involve a water bath, you simply fill the jars, put the lid and band on them, and turn them upside down. There's question on how safe that is. (I know you didn't do the inversion method, but it's close even if by accident.)
I know marmalades can take up to two weeks to firm up and if they don't by then, it's suggested to reprocess them. At this point, I would say you would be safe to reprocess them. I wouldn't put what you have directly into a water bath though. I would pour the contents back into a pot and bring to a boil. Meanwhile, wash the jars and bands and use new lids. Then fill the clean (hot) jars and put into a water bath.