Indeed this is a subject of many legends and outlandish ideas. Possibly because of vague views on what exactly hands-free means. Freedivers mostly call it hands free or successful hands-free when there's no hand on the nose. There is no hand on nose means that there is hands-free equalization. On one hand this is undoubtedly true. On the other hand, there are subtleties. Let us dig it.
Once I was delivering a talk to young scuba divers on equalization. After introducing a list of ways to equalize, I asked the kids how they do it. To my big surprise, most did it hands-free by swallowing. Sure with scuba they descend in a horizontal position, sure they can raise the head for ease, sure they have plenty of air. But they don't pinch the nose, so their method can be called hands-free. In freediving though such swallowing hands-free is not widespread. Rarely freedivers report using it. On youtube many freedivers demonstrate their skills of equalization without pinching the nose. If the angle of the video and quality are o.k., we can often see bubbles escaping from under the mask at regular intervals. This is a clue of pushing air up into the nasal cavity during the hands-free equalization. Small movements of the tongue, larynx, cheeks do create extra air pressure. The more experienced a freediver is, the fewer bubbles escape, the better is his or her technique. This option of the hands-free is differentiated from Frenzel-Fattah only by a lesser effort, while the muscles involved are the same. We have small muscles for opening the auditory tubes. These tubes connect the nasal cavity to the inner ears. By default the tubes are closed, their section flattened, and the air can not flow into the ears to equalize. Would be great to learn to operate these tube-opener muscles called "tensor veli" at will, in order to equalize without pushing air up, nor swallowing. The tubes would just open and let the air in the ears. This would be really hands-free, plus requiring minimum effort and be effective at any depth, as long as there is some spare air in the larynx and mouth. Let us look into the anatomy. I love this cutout picture allowing to see a detailed view on how the "tensor veli" tube-opening muscle opens it. The muscles evolved to sideways stretch the soft palate. This muscle runs sideways from the soft palate, uses a bony hook as a pulley to turn up, goes to an entire pinched section of the auditory tube and while tensed it pulls it down, opening it for airflow.
The next picture shows it from inside, as if we'd look from inside the head towards the nasal cavity. We can't see the tube cutout, just it's orifice. Here too we see the muscle "tensor veli" (#5) also I'd call it kinda palate-stretcherer, it goes down from the tube orifice, and around the pulley-bony-hook, and to the soft palate. And we see some other muscles around the soft palate as an intricate muscular structure, all interlinked. Below we will mention two more pairs of muscles from this picture: #4 "levator veli" elevating the soft palate thus bypassing the pulley-bony-hook nor really pulling the entire tube to open, and #3 "palatopharyngeus" pulling the soft palate downwards.
Next picture explains in 3D the soft palate (and the attachment of the palate-stretcherer to it).
Next picture shows again an interconnection of the muscles of the soft palate and the entrance to the auditory tube.
Finally a picture demonstrating a complicated web formed by a big number of different muscles there, thus giving the reason why separating the ones we need to control equalization from this mess is pretty difficult.
Thus from anatomy we can single out a pair of tube-opener muscles stretching the soft-palate sideways and eventually two more pairs of muscles controlling the soft palate up and down (detailed below). Those are the ones to master for hands-free equalization. Only they do open the auditory tubes and let air to flow into the ears. No other muscle from the throat or tongue or cheeks would actually open the tubes.
Back to practical solutions. How to learn opening the tubes? One way is going backwards capitalizing on innate automated movements. Deconstructing carefully our smallest inner perceptions when we swallow or yawn. First you hear clicks, claps, crunching, champing, or some other personal thing. It means the tubes open, at least briefly. Then you try to feel the muscles involved, how the palate moves, etc. And try to learn using tube-openers/palate-stretchers only and deliberately. One trick is to put earbuds into the nostrils and put the volume on. It will remain muffled by all the bones and soft tissues inside the head. Only when the tubes open, the sound would become louder, as it would clearly reach the inner ears through the open tubes. The goal is to open the tubes and thus to hear the sound clearly for a minute or so. After learning that, continue to the water, diving very slowly, fine-tuning the hands-free opening of the tubes. Another way is to go straightforward, starting at learning to move the soft palate, contracting and relaxing it, and progress towards tube-opening. The palate opens a passage from nose to mouth automatically when breathing through the nose, and it closes this mouth-nose passage when swallowing (to avoid letting food or liquid go up into the nasal cavity). The problem is there is the tongue moving synchronized with the soft-palate in the swallowing action. We need to learn to disconnect the palate from the tongue. Practice to inhale through the nose, exhale through the mouth, and the other way around. Feel the palate. Learn to move it up and down deliberately, closing and opening the nose-to-mouth passage. On the first short video muscles move the palate up. These muscles "levator veli" alone will not open the tubes. Even worse, when the palate is completely up, air can't even reach the nasal cavity where the orifices of auditory tubes are. And as a dive progresses, the closed palate will be sucked-up by a growing decompression in the nasal cavity, making it virtually impossible to open it up again, at least without swallowing the air stored in the throat and cheeks.
After learning to move the palate up, you'd add control of another pair of muscles, those pulling the palate down and called "palatopharyngeus". These muscles are connecting the palate somewhere to the larynx and passing near the base of the tongue. When these muscles are tense, elevating the palate would be counteracting them. It'll feel like effort to elevate the palate. This time palate-elevating muscles may involuntarily involve palate-stretching muscles and actually pull the wall of the tubes down, somehow opening the tubes for equalization. Or they may not.
This time on each side of the throat you see downwards pulling "palatopharyngeus" muscles becoming salient. They are somehow closing towards each other. Try looking in a mirror or use a camera. The more these muscles would be closing together, the more they'd pull the palate down, the more effort you'd need to keep it up with palate elevators, the more pull the palate elevators will apply on the tubes, facilitating opening those, especially when combined with blowing air into a pinched nose (which is not hands-free). And also the downward pulling muscles will not allow the palate-elevators to completely lift the palate thus not obturating air passage to the nasal cavity. Air would flow into it and further into more or less opened auditory tubes in the ears.
The trick with earbuds in the nostrils will help at this stage also. The ultimate thing is finally to use muscles stretching the soft palate sideways without involving palate-levators neither palate-depressor muscles. In medical terms: using only "tensor veli" without involving "levator veli" nor "palatopharyngeus". On the third short video we try to show in a succession the action of pulling the palate up, adding down-pullers of the palate, and only leaving palate-stretchers tensed for tubes to remain opened.
Since we freedive head down, the soft palate is additionally pulled by gravity towards obturating the nasal cavity. Keeping the palate tense by palate-stretchers will prevent this (unless you have an unusually long soft palate. In such a case, an additional tension of a 4th pair of muscles "palatoglossus", please see furthermore part 3 of this post). Very useful to avoid having the nasal cavity becoming obturated by gradually increasing suction of lower air pressure in the nose, usually leading to equalization failure.