Calcite, the most stable polymorph of calcium carbonate, is one of the most abundant simple salts in the geological environment. Consequently, its natural (10 (1) over bar4) cleavage plane has been studied extensively by a wide range of surface-sensitive techniques, giving indications for two reconstructions, namely a (2 x 1) and a so-called 'row-pairing' reconstruction. The existence of the (2 x 1) reconstruction has been discussed controversially in the literature, but is now confirmed as a true surface property. In contrast, a comprehensive discussion on the existence of the row-pairing reconstruction is lacking so far. Here, we present a non-contact atomic force microscopy (NC-AFM) study of the (10 (1) over bar4) calcite surface performed in an ultra-high vacuum. We discuss a broad variety of different NC-AFM contrasts and present a comprehensive classification scheme. This scheme encompasses a total of 12 different contrast modes. Atomically resolved NC-AFM images are shown, giving experimental evidence for 10 of these contrast modes. In particular, some of these modes allow for identification of the two surface reconstructions while others do not. This variety in appearances provides an explanation for the seemingly contradicting observations in the literature. Based on a detailed investigation of the influence of tip termination and interaction regime, we further analyse the existence of the row-pairing reconstruction.