Walking on their hands is certainly the easiest explanation on the face of it. You could simply say that's how they do it and leave it at that; it's fairly logical and quite suitable if you don't want to get into it.
However, if you want to be precise, there are some issues with that. While some people are obviously very good at walking on their hands, it takes a lot of practice and a lot of balance. I, for example, cannot walk on my hands, nor do I need to. The average mermaid, as you said, lives in water, eats in water, and has no need to go on land. Unless your mermaids are habitually traveling over land, there is no reason for them to have the adaptations necessary to walk on their hands. Besides the adaptations mentioned in regards to head rotation and elbow position, extensive shoulder modification would be necessary for habitual hand-walking. The human pelvis is highly specialized for bipedal walking, including proper support and carriage of internal organs. The human body is not designed to be inverted for long periods or on a regular basis, which doesn't mean that it can't be, but it does mean that you need a good reason for the mermaids to go that route. See these pages for info on the human pelvis and on the risks of being inverted (just hanging, not including joint issues from reversed gravity)- https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5545133/ https://www.healthline.com/health/hanging-upside-down#whats-the-max
The other problem with hand-walking is in terrain. There is a reason that no land animals have their heads between their legs. If your mermaids have human proportions, their heads will hang to about the level of their elbows when walking on their hands, meaning that even something like stairs would be challenging for them simply because their heads are in the way. If your mermaids are simply walking down the road, that's not much of an issue, but if they live in semi-wild areas, they're going to have trouble. Living in the water, they are also going to face much more sand, mud, rock, and marsh than most people do. Even with our advanced balance (which I'll come back to), our long legs, and our high vantage point, humans generally avoid traversing loose sand, slick mud and rock, and deep marsh because it's a pain in the behind to get through. For a mermaid walking on her hands, these terrains would be incredibly difficult, both because she can't well see what's coming up or how to navigate it and because her head is already in danger of being whacked on the topography or lost below the surface.
Balance is another factor in both rough and fair terrain. Humans, besides having specially evolved features to assist in bipedal locomotion, have arms we can windmill wildly to help us keep our balance. Even when we walk on our hands, we can use our legs to help steady us. A mermaid cannot. Even if she has incredible balance from years of practice, she's going to have a terrible time remaining steady on anything but a flat road. Additionally, because she doesn't have independent legs, she'll have to have abs of absolute steel just to get herself upright in the first place. Humans, when getting up on their hands, will usually swing one leg up first, to create the momentum to lift their other leg into position. Unless a mermaid has assistance, she's going to have to lift her entire body off the ground in a plank position and then raise it with the sheer force of muscle power, or do some kind of worm dance to heave her back half up into position. Not impossible, but again, not a first-time sort of thing.
If you want a realistic, effective, biological means of land locomotion for your mermaids, the best way to go about it is probably to modify them to account for all these limiting factors and just go with hand-stands. It will require physiological alterations that would have taken a very, very long time to develop (and so must have been selected for over millennia) and which might change the appearance of your creatures though. Since, however, this is a work of fiction, you could simply ignore or hand-wave any or all of these concerns.
I think the most realistic way that mermaids would move over land, given that they should have no regular need to, would be for them to slither like eels and octopods, and would do so for roughly the same reasons. If, for example, a mermaid got caught in a tide pool or needs to get from one lake into another. It would be slow and probably ungraceful, but it would also work better in the areas it is likely to be necessary. While hand-walking doesn't work well in wet sand and mud, belly-slithering is perfect for those conditions.
In a world with mermaids, it's also very possible that those mermaids have some kind of magic or siren song that would make outside intervention possible. Maybe they cloud men's minds to do their bidding, carrying them across great distances. Maybe they have an ancient kinship with another mythical beast like the centaur, the unicorn, or the dragon, which they are able to leverage for aid. Maybe they have a mystical connection with wild beasts and can call on them in times of need. For a semi-biological explanation, the mermaids could have a sort of tranquilizing venom they use to stun large animals like moose, bison, or horses, then drag themselves up onto the backs of these stupefied beasts and ride them like zombies. Heck, maybe they have giant frogs as pets and can ride them about whenever they feel like it, or have sleighs pulled by crocodiles.
In terms of a classical mermaid- ie, one without wings, without four legs, and with a single-column tail- there just isn't a good way for them to move on land. That's why there aren't myths about mermaids on land. If you are willing to modify the basic mermaid body, or if you are willing to simply say "magic" and move on, the options are limitless.
