Polarization profiles are presented for 20 millisecond pulsars that are being observed as part of the Parkes Pulsar Timing Array project. The observations used the Parkes multibeam receiver with a central frequency of 1369 MHz and the Parkes digital filterbank pulsar signal-processing system PDFB2. Because of the large total observing time, the summed polarization profiles have very high signal/noise ratios and show many previously undetected profile features. Thirteen of the 20 pulsars show emission over more than half of the pulse period. Polarization variations across the profiles are complex and the observed position angle variations are generally not in accord with the rotating-vector model for pulsar polarization. Never-the-less, the polarization properties are broadly similar to those of normal (non-millisecond) pulsars, suggesting that the basic radio emission mechanism is the same in both classes of pulsar. The results support the idea that radio emission from millisecond pulsars originates high in the pulsar magnetosphere, probably close to the emission regions for high-energy X-ray and gamma-ray emission. Rotation measures were obtained for all 20 pulsars, eight of which had no previously published measurements. ; Comment: 15pages, 21 figures, 3 tables, accepted for publication in MNRAS