We examine the potential of asteroseismology for exploring the internal rotation of white dwarf stars. Data from global observing campaigns have revealed a wealth of frequencies, some of which show the signature of rotational splitting. Tools developed for helioseismology to use many solar p-mode frequencies for inversion of the rotation rate with depth are adapted to the case of more limited numbers of modes of low degree. We find that the small number of available modes in white dwarfs, coupled with the similarity between the rotational-splitting kernels of the modes, renders direct inversion unstable. Accordingly, we adopt what we consider to be plausible functional forms for the differential rotation profile; this is sufficiently restrictive to enable us to carry out a useful calibration. We show examples of this technique for PG 1159 stars and pulsating DB white dwarfs. Published frequency splittings for white dwarfs are currently not accurate enough for meaningful inversions; reanalysis of existing data can provide splittings of sufficient accuracy when the frequencies of individual peaks are extracted via least-squares fitting or multipeak decompositions. We find that when mode trapping is evident in the period spacing of g modes, the measured splittings can constrain dOmega/dr. ; Comment: 26 pages, 20 postscript figures. Accepted for publication in The Astrophysical Journal