We present an analysis of CO spectroscopy and infrared-to-millimetre dust photometry of 11 exceptionally bright far-infrared (FIR) and sub-mm sources discovered through a combination of the Planck all-sky survey and follow-up Herschel-SPIRE imaging -"Planck's Dusty Gravitationally Enhanced subMillimetre Sources". Each source has a secure spectroscopic redshift z = 2.2-3.6 from multiple lines obtained through a blind redshift search with EMIR at the IRAM 30-m telescope. Interferometry was obtained at IRAM and the SMA, and along with optical/near-infrared imaging obtained at the CFHT and the VLT reveal morphologies consistent with strongly gravitationally lensed sources, including several giant arcs. Additional photometry was obtained with JCMT/SCUBA-2 and IRAM/GISMO at 850 µm and 2 mm, respectively. The SEDs of our sources peak near either the 350 µm or 500 µm bands of SPIRE with peak flux densities between 0.35 and 1.14 Jy. All objects are extremely bright isolated point sources in the 18 beam of SPIRE at 250 µm, with apparent FIR luminosities of up to 3 × 10 14 L (not correcting for the lensing effect). Their morphologies, sizes, CO line widths, CO luminosities, dust temperatures, and FIR luminosities provide additional empirical evidence that these are amongst the brightest strongly gravitationally lensed high-redshift galaxies on the sub-mm sky. Our programme extends the successful wide-area searches for strongly gravitationally lensed high-redshift galaxies (carried out with the South Pole Telescope and Herschel) towards even brighter sources, which are so rare that their systematic identification requires a genuine all-sky survey like Planck. Six sources are above the 600 mJy 90% completeness limit of the Planck catalogue of compact sources (PCCS) at 545 and 857 GHz, which implies that these must literally be amongst the brightest high-redshift FIR and sub-mm sources on the extragalactic sky. We discuss their dust masses and temperatures, and use additional WISE 22-µm photometry and template fitting to rule out a significant contribution of AGN heating to the total infrared luminosity. Six sources are detected in FIRST at 1.4 GHz, and the others have sensitive upper limits. Four have flux densities brighter than expected from the local FIR-radio correlation, but in the range previously found for high-z sub-mm galaxies, one has a deficit of FIR emission, and 6 are consistent with the local correlation, although this includes 3 galaxies with upper limits. We attribute this to the turbulent interstellar medium of these galaxies, rather than the presence of radio AGN. The global dust-to-gas ratios and star-formation efficiencies of our sources are predominantly in the range expected from massive, metal-rich, intense, high-redshift starbursts. An extensive multi-wavelength follow-up programme is being carried out to further characterize these sources and the intense star formation within them.Key words. galaxies: high-redshift -galaxies: star formation -galaxies: starburst -submillimeter: galaxies -gravi...