Many traditional methods for analyzing gene-gene relationships focus on positive and negative correlations, both of which are a kind of 'symmetric' relationship. However, genes can also exhibit 'asymmetric' relationships, such as 'if-then' relationships used in boolean circuits. In this paper we develop a very general method that can be used to detect biclusters within gene-expression data that involve subsets of genes which are enriched for these 'boolean-asymmetric' relationships (BARs). These BAR-biclusters can correspond to heterogeneity that is driven by asymmetric gene-gene interactions, rather than more standard symmetric interactions. We apply our method to a single-cell RNA-sequencing data-set, demonstrating that the statistically-significant BAR-biclusters indeed contain additional information not present within more traditional 'boolean-symmetric'-biclusters. For example, the BAR-biclusters involve different subsets of cells, and highlight different gene-pathways within the data-set. Moreover, by combining the boolean-asymmetric- and boolean-symmetric-signals, one can build linear classifiers which outperform those built using only traditional boolean-symmetric signals.