We propose that the ``6 s'' pulsars, as well as other possible evolutionary analogs of one of these, namely, 4U 1626-67, are products of common envelope (CE) evolution of high-mass X-ray binaries (HMXBs). Close HMXBs produce Thorne-Żytkov objects due to complete spiral-in during the CE phase, which turn either into black hole systems or into 6 s pulsars like 4U 0142+61 and 1E 2259+586 (van Paradijs, Taam, & van den Heuvel), which we call braking X-ray pulsars (BXPs). Relatively wide HMXBs eject the massive companion's envelope during the CE phase, producing helium star-neutron star binaries (HSNBs) like 4U 1626-67, Cyg X-3, and perhaps the newly discovered X-ray binary HD 49798. We show that the accretion flow in BXPs is likely to have a significant spherical component which would lead to soft X-ray spectra and large equivalent blackbody areas, in accordance with observations. We argue that, on theoretical grounds, HSNBs like 4U 1626-67 are expected to show both spin-up and spin-down, and that this property sets them apart from BXPs, which are expected to show only spin-down. We discuss the subsequent evolution of BXPs and HSNBs, and stress their importance as missing links in the zoo of neutron star binaries.