What is horizontal about a horizontal symmetry?

Question

When studying a seminal paper in flavor physics, Mass Matrix Models (released in 1992), I came across the term, a horizontal symmetry. Googling around I saw this was (is?) indeed a commonly used term. They appear to define this symmetry in the context of the Standard Model as, \begin{equation} Q \rightarrow L Q , \bar{d} \rightarrow R _d \bar{d} , \bar{u} \rightarrow R _u \bar{u} , \phi \rightarrow P \phi \end{equation} where $L,R_u,R_d$ mix between the generations of quarks and $P$ mixes between the color singlet doublet scalars in the model.

It just seems to be the most general flavor symmetry assuming $3$ generations and consistent with gauge symmetry. Is there something that I am missing about the meaning of a horizontal symmetry and in what sense is it "horizontal"?

Answer 1

The textbook An Introductory Course of Particle Physics by Palash B. Pal offers a (slightly unsatisfactory) answer. The author writes that

Such symmetris are often called horizontal symmetries. The name is a pictorial reminder to the list of quark fields in Eq. 17.1: transformations in this group act horizontally in this arrangement.

I don't have the book and the page with Eq. 17.1 isn't avaiable in the preview, but I imagine it's simply a horizontal list of quarks e.g. $u$, $c$, $t$. I find this quite tenuous, and wonder whether there is another deeper reason.