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In the 1920s, during the first archaeological excavations at Uaxactún, Petén, Guatemala, an architectural complex named Group E was interpreted as an ancient Maya astronomical observatory, intended specifically for sighting the equinoctial and solstitial sunrises. In the following decades, a large number of architectural compounds with the same configuration have been found, most of them in the central lowlands of the Yucatan peninsula. The multiple hypotheses that have been proposed about the astronomical function of these complexes, commonly designated as E Groups, range from those attributing them a paramount role in astronomical observations to those that consider them merely allegorical or commemorative allusions to celestial cycles, without any observational use. This study, based on quantitative analyses of a reasonably large sample of alignment data, as well as on contextual evidence, shows that many of the previous hypotheses cannot be sustained. I argue that E Groups, although built primarily for ritual purposes, were astronomically functional, but also that they had no specific or particularly prominent role in astronomical observations. Their orientations belong to widespread alignment groups, mostly materialized in buildings of other types and explicable in terms of some fundamental concerns of the agriculturally-based Maya societies. I present the evidence demonstrating that the astronomical orientations initially embedded in E Groups, which represent the earliest standardized form of Maya monumental architecture and whose occurrence in practically all early cities in the central Yucatan peninsula attests to their socio-political significance, were later transferred to buildings and compounds of other types. Therefore, it is precisely the importance of the astronomically and cosmologically significant directions, first incorporated in E Groups, that allows us to understand some prominent aspects of ancient Maya architecture and urbanism.