We derive the spectral types of the cool giants in about 100 symbiotic systems. Our classification is mainly based on near IR spectra in order to avoid the contamination of the spectrum by the nebula and the hot component in the visual region. The accuracy of our spectral types is approximately one spectral subclass, similar to previous near IR classification work, and much better than visual spectral type estimates. Strong, intrinsic spectral type variations (>2 spectral subtypes) are only seen in systems containing pulsating mira variables. We present a catalogue of spectral types for cool giants in symbiotic systems which also includes determinations taken from the literature. The catalogue gives spectral types for the cool giants in about 170 systems which is nearly the full set of confirmed symbiotics. Based on our classifications we discuss the distribution of spectral types of the cool giants in galactic symbiotic binaries. We find that the spectral types cluster strongly between M3 and M6, with a peak at M5. The distribution of systems with a mira variable component peaks even later, at spectral types M6 and M7. This is a strong bias towards late spectral types when compared to red giants in the solar neighbourhood. Also the frequency of mira variables is much larger among symbiotic giants. This predominance of very late M-giants in symbiotic systems seems to indicate that large mass loss is a key ingredient for triggering symbiotic activity on a white dwarf companion. Further we find for symbiotic systems a strong correlation between the spectral type of the cool giant and the orbital period. In particular we find a tight relation for the minimum orbital period for symbiotic systems with red giants of a given spectral type. This limiting line in the spectral type - orbital period diagram seems to be equivalent with the relation R<=l_1/2, where R is the radius of the red giant and l_1 the distance from the center of the giant to the inner Lagrangian point L_1. This correlation possibly discloses that symbiotic stars are - with probably only one exception in our sample - well detached binary systems. Based on observations obtained with the 1.52~m and 3.6~m telescopes of the European Southern Observatory (ESO), the 1.93~m telescope of the Observatoire de Haute-Provence (OHP), the 2.3~m telescope of the Australian National University (ANU) at Siding Spring, and the William Herschel Telescope (WHT) at La Palma. This research has made use of the AFOEV database, operated at CDS, France.