The notion of inverse M-matrices has been quite well studied and many of their properties have been documented in the literature. In this article, a proposal to consider the larger class of inverse H-matrices, is made. First, certain special types of real matrices are identified as inverse H-matrices. While these matrices are not inverse M-matrices in general, their definitions are inspired by matrices that are known to be inverse M-matrices. In the rest of the article, motivated by results that hold for inverse M-matrices, the endeavor will be to attempt at proving corresponding results for inverse H-matrices. It is shown that for the full class of inverse H-matrices, many of these extensions are false. Nevertheless, two subclasses of real inverse H-matrices are identified, where these generalizations are shown to hold. © 2019 Elsevier Inc.