I don't know how the Recoil program computes a skewed Lorentzian, but I can tell you how it is implemented in my analysis programs.
A "skewed Lorentzian" consists, according to the way I defined it for our work on nanophase hematites (ref. below), of two half Lorentzians with different widths, joined at the left-right mid-line so that their maximum heights agree. If W = FWHM and sk = the "skew" parameter, then the Left Half-Width of the Lorentzian = W (1 - sk) / 2, while the Right Half-Width = W (1 + sk) / 2. In this way W = FWHM is maintained also for the skewed Lorentzian.
Reference:
Evidence for pigmentary hematite on Mars based on optical, magnetic, and Mossbauer studies of superparamagnetic (nanocrystalline) hematite, by Richard V. Morris, David G. Agresti, Howard V. Lauer Jr., Jeffery A. Newcomb, Tad D. Shelfer, A. V. Murali, in Journal of Geophysical Research (1989) Vol 94, 2760-2778.