The notion of type of quadruples of rows is proven to be useful in the classification of Hadamard matrices. We investigate Hadamard matrices with few distinct types. Apparently, Hadamard matrices with few distinct types are very rare and have nice combinatorial properties. We show that there exists no Hadamard matrix of order larger than $12$ whose quadruples of rows are all of the same type. We then focus on Hadamard metrics with two distinct types. Among other results, the Sylvester Hadamard matrices are shown to be characterized by their spectrum of types. This is a joint work with A. Mohammadian.