The least-squares refinement of non-centrosymmetric crystal structures as inversion twins is presented. It is shown that the absolute-structure (twin) parameter x may be used to define the chirality or polarity of untwinned crystals. The method has been applied to 20 compounds. The least-squares refinement of the absolute-structure parameter is rapid and stable. The value of x generally falls within three e.s.d.'s of the physically meaningful range 0-< x -1 and the e.s.d.'s increase as f' becomes smaller. New residual and goodness-of-fit values are defined to judge the efficiency of the method. The estimated standard deviation of x, taken with a pseudo Durbin-Watson d statistic, provides an excellent criterion for the reliability of the absolute-structure determination. Refinements on data sets including very accurately measured Friedel pairs of reflections have also been tested. The determination of the free direction(s) of origin-free space groups and an efficient algorithm for the inversion of a crystal structure that refines to x-~ 1 are given in detail. The data and procedural structures necessary for an efficient computer implementation of absolute-structure refinement are also considered. The formulae giving the correction for the effects of anomalous dispersion on IFobsI from an inversiontwinned crystal are given. These corrected Fobs are the ones to be used in an electron-density calculation. The correlation of residuals following least-squares refinement is quantified by using a pseudo DurbinWatson d statistic. The causes of the correlation, its effect on the value of x and its e.s.d., and ways of avoiding the correlation are considered. It is shown that in using x it is more suitable to refine on IFI 2 than IF[. A weighting scheme is presented and tested that increases the sensitivity of a refinement to absolute structure.0108-7673/85/050500-12501.50