Boolean matrix factorization (BMF), a popular methodology of preprocessing and analyzing 1/0 tabular data, generally handles Os and Is differently. It aims to explain Is in the data by factors, while Os are just left unexplained. This difference is mainly given by the usual data character, where 1s carry much more important information (and are much scarcer) than Os. However, in some datasets, the Is and Os are equally important. Such datasets require symmetrical handling of Is and Os. We propose a novel factorization of such data and its algorithm. Unlike usual BMF methods, factors are linearly ordered by priority in our factorization, and factors can contradict each other – meaning that one factor can put 1 where the other puts 0. In such a case, the factor with higher priority is right. We show that the proposed factorization provides a more compact data description than a straightforward application of the usual BMF methods.