Two-dimensional arrays of ferroelectric lead zirconate titanate (PZT) nanodots were fabricated using pulsed laser deposition through ultrathin anodic aluminum oxide membrane stencil masks. The static distribution of polarization configurations was investigated using in- and out-of-plane piezoresponse force microscopy (PFM). The observed presence of an in-plane polarization component in nominally (001) oriented PZT suggests the existence of a significant deviation from the regular tetragonal structure that allows the formation of complex core-polarization states. Core-polarization states may indicate the presence of quasi-toroidal polarization ordering. The experimental results are compared with a theoretical model to determine the fingerprint of a vortex polarization state in PFM.