Hearts in which the arterial trunks arise from the morphologically appropriate ventricles, but in a parallel manner, rather than the usual spiralling arrangement, have long fascinated anatomists. These rare entities, for quite some time, were considered embryological impossibilities, but ongoing experience has shown that they can be found in various segmental combinations. Problems still exist about how best to describe them, as the different variants are often described with esoteric terms, such as anatomically corrected malposition or isolated ventricular inversion. In this review, based on our combined clinical and morphological experience, we demonstrate that the essential feature of all hearts described in this manner is a parallel arrangement of the arterial trunks as they exit from the ventricular mass. We show that the relationship of the arterial roots needs to be described in terms of the underlying ventricular topology, rather than according to the arrangement of the atrial chambers. We then discuss the importance of determining atrial arrangement on the basis of the morphology of the appendages, following the precepts as set out in the so-called "morphological method" and distinguished according to the extent of the pectinate muscles relative to the atrioventricular junctions as opposed to basing diagnosis on the venoatrial connections. We show that, when approached in this manner, the various combinations can be readily diagnosed in the clinical setting and described in straightforward way.