After several decades I have figured out that there is a way of brushing teeth that is probably better than the traditional ones, all variants of holding the tooth-brush horizontally: brushing sideways, up-and-down, with a circular motion.
That is to hold and move the brush vertically, which is probably better because:
* Most cavities happen between teeth, rather than on their inner or outer surfaces, because food gets more easily trapped or stuck and decays there, and holding the brush vertically and moving it vertically aligns it better with the gaps between teeth.
* A lot of teeth problems come from the interface between tooth and gum, and holding the brush vertically means brushing across the length rather than the width of the head and thus increases the duration of the brushing of that interface, because the heads is longer that way.
* Most importantly, when holding the brush horizontally it is very awkward to brush the gaps on inner side of the teeth, while it is much easier, as both the head is narrower and the handle is unobstructed by the other teeth, when holding the brush vertically, head high for the upper teeth, head down for the lower teeth.
There is a case for holding the tooth-brush horizontally during the brushing of the crown side of the teeth, which is useful because sometimes food gets stuck there too, but there is also a case for doing it by brushing across the length rather than the width of the head.