A cursory search reveals that Latin squares are fairly extensively used in the design of experiments. During my PhD, I have studied various theoretical properties of Latin squares (from a combinatorics point-of-view), but do not have a deep understanding of what is it about Latin squares that make them particularly well-suited to experimental design.
I understand that Latin squares are good at allowing statisticians to efficiently study situations where there are two factors which vary in different "directions". But, I'm also fairly confident there would be many other techniques that could be used.
What is it, in particular, about Latin squares that make them so well suited for the design of experiments, that other designs do not have?
Moreover, there are zillions of Latin squares to choose from, so which Latin square do you choose? I understand that choosing one at random is important, but there would still be some Latin squares that would be less suited to running experiments than others (e.g. the Cayley table of a cyclic group). This raises the following question.
Which properties of Latin squares are desirable and which properties of Latin squares are undesirable for experimental design?