We outline the theory and practice of measuring the four Stokes parameters of spectral lines in emission/absorption observations. We apply these concepts to our Arecibo H I absorption line data and present the results. We include a detailed discussion of instrumental effects arising from polarized beam structure and its interaction with the spatially extended emission line structure. At Arecibo, linear polarization [Stokes (Q,U)] has much larger instrumental effects than circular (Stokes V). We show how to reduce the instrumental contributions to V and to evaluate upper limits to its remaining instrumental errors by using the (Q,U) results. These efforts work well for opacity spectra but not for emission spectra. Arecibo's large central blockage exacerbates these effects, particularly for emission profiles, and other telescopes with weaker sidelobes are not as susceptible. We present graphical results for 41 sources; we analyze these absorption spectra in terms of Gaussian components, which number 136, and present physical parameters including magnetic field for each.