We investigate the stellar angular momentum of galaxies across the Hubble sequence from the CALIFA survey. The distribution of CALIFA elliptical and lenticular galaxies in the λRe - ɛe diagram is consistent with that shown by the Atlas3D survey. Our data, however, show that the location of spiral galaxies in this diagram is significantly different. We have found two families of spiral galaxies with particularly peculiar properties: (a) spiral galaxies with much higher λRe values than any elliptical and lenticular galaxy; (b) low-mass spiral galaxies with observed λRe values much lower than expected for their apparent flattening. We use these two families of objects to argue that (1) fading alone cannot explain the transformation of spiral to lenticular galaxies, and (2) that those low-mass spiral galaxies are in fact dark matter dominated, which explains the unusually low angular momentum.