The problem of misclassification is common in epidemiological and clinical research. In some cases, misclassification may be incurred when measuring both exposure and outcome variables. It is well known that validity of analytic results (e.g. point and confidence interval estimates for odds ratios of interest) can be forfeited when no correction effort is made. Therefore, valid and accessible methods with which to deal with these issues remain in high demand. Here, we elucidate extensions of well-studied methods in order to facilitate misclassification adjustment when a binary outcome and binary exposure variable are both subject to misclassification. By formulating generalizations of assumptions underlying well-studied "matrix" and "inverse matrix" methods into the framework of maximum likelihood, our approach allows the flexible modeling of a richer set of misclassification mechanisms when adequate internal validation data are available. The value of our extensions and a strong case for the internal validation design are demonstrated by means of simulations and analysis of bacterial vaginosis and trichomoniasis data from the HIV Epidemiology Research Study.