A fraction of active galactic nuclei (AGN) have double-peaked H, H and Mg II broad lines attributed to emission from rotating gas in the accretion disk. Using optical spectroscopy of a flux-limited sample of AGN selected via ultrahard X-rays from the BAT AGN Spectroscopic Survey (BASS), we systematically identify 71 double-peaked emitters amongst 343 broad-line AGN with redshifts and 2-10 KeV X-ray luminosities of log 40-45.7 (erg/s), and provide their best-fit accretion disk geometry parameters. We find that ~21% of X-ray selected broad-line AGN are double-peaked emitters (DPEs), consistent with rates previously reported for broad-line AGN selected for strong optical variability in ZTF. 11 of 71 DPEs (15%) exhibited a single-peaked Gaussian component to the broad line profile in addition to the double-peaked disk profile. In this sample, DPEs have intrinsically higher masses by ~0.4 dex and lower Eddington ratios by ~0.3 dex than other broad-line AGN, and have a preference for elliptical host galaxies, higher X-ray luminosities, and higher [OI] 6302 to narrow H flux ratios than other broad-line AGN. We find that DPEs are not segregated from other broad-line AGN in the vs relation or their X-ray to radio luminosity ratios, and do not show a preference for intermediate Seyfert types over Seyfert 1s. We do not find differences in a wide range of multi-wavelength properties when comparing DPEs to other broad-line AGN, including optical and mid-IR variability levels, and the rate of changing-look events. We discuss the two populations in the context of multi-component disk-wind models of the AGN broad line region and consider how unrecognized contributions of disk emission to the broad lines introduce biases in virial SMBH mass estimates.

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