We study finite set-theoretic solutions $(X,r)$ of the Yang-Baxter equation of square-free multipermutation type. We show that each such solution over $\C$ with multipermutation level two can be put in diagonal form with the associated Yang-Baxter algebra $A=A(\C,X,r)$ having a $q$-commutation form of relations determined by complex phase factors. These complex factors are roots of unity and all roots of a prescribed form appear as determined by the representation theory of the finite abelian group $\Gcal$ of left actions on $X$. We study the structure of the algebras $A(\C,X,r)$ and show that they have a $\bullet$-product form `quantizing' the commutative algebra of polynomials in $|X|$ variables. We obtain the $\bullet$-product both as a Drinfeld cotwist for a certain canonical 2-cocycle and as a braided-opposite product for a certain crossed $\Gcal$-module (over any field $k$). We provide first steps in the noncommutative differential geometry of $\Acal(k,X,r)$ arising from these results.