Two modifications of CeCuSn were prepared from the elements: the high-temperature (beta) modification crystallizes directly from the quenched sample, while the low-temperature (alpha) modification forms after annealing at 700 deg C for one month. Both modifications were investigated by X-ray powder and single crystal diffraction. We find for alpha-CeCuSn a structure of ZrBeSi type, space group P63/mmc, a = 458.2(1), c = 793.7(2) pm, wR2 = 0.0727, 148 F2 values, 8 variable parameters. In the case of beta-CeCuSn we find the NdPtSb type structure, space group P63mc, a = 458.4(1), c = 785.8(2) pm, wR2 = 0.0764, 233 F2 values, 11 variable parameters. The copper and tin atoms build up layers of ordered [Cu3Sn3] hexagons. The layers are planar in beta-CeCuSn, however, with highly anisotropic displacements of the copper and tin atoms. In alpha-CeCuSn a puckering effect is observed resulting in a decrease of the c lattice parameter. Both modifications of CeCuSn exhibit antiferromagnetic ordering, however, there is a considerable difference in their magnetic behaviour. We show the anomalies in the physical properties of the alpha- and beta- modifications of CeCuSn by Mossbauer spectroscopy,magnetic and specific heat measurements and explain their structure-property relations.