Spin polarization transfer from parahydrogen (p-H(2)) to another molecular entity is generally thought to be mediated by longitudinal spin order (represented by the operator product I(z)(A)I(z)(B), A and B being the two hydrogen nuclei which originate from p-H(2) after a hydrogenation reaction). The longitudinal spin order leads to antiphase patterns in the proton NMR spectrum. In addition to these antiphase patterns, in-phase patterns, arising from polarization differences (represented by (I(z)(A)-I(z)(B))), have been experimentally observed. A complete theory, based on a density operator treatment, has been worked out and applied to the two types of parahydrogen induced polarization experiments: PASADENA (PArahydrogen and Synthesis Allow Dramatically Enhanced Nuclear Alignment; hydrogenation reaction inside the NMR magnet) and (ALTADENA) (Adiabatic Longitudinal Transport After Dissociation Engenders Nuclear Alignment; hydrogenation reaction outside the NMR magnet). It is shown that polarization differences are always created in the case of a PASADENA experiment but that their amplitude depends critically on the ratio of the J coupling over the frequency difference between A and B. In the case of an ALTADENA experiment, if the sample is slowly transferred toward the NMR magnet, polarization differences are definitely created and their amplitude can be larger than the amplitude of the longitudinal spin order. Some test experiments demonstrate the validity of the proposed theory.